Fractional Statistics: a Retrospective View

Home , Grand canonical ensemble, Parastatistics

Fractional statistics: A retrospective view

Andrij Rovenchak

Department for Theoretical Physics, Ivan Franko National University of Lviv

International School and Workshop Anyon Physics of Ultracold Atomic Gases 12–15 December 2014

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 1 / 60 Talk outline

1 Introduction 2 Quantum-mechanical approach Anyons q-deformations Parastatistics 3 Statistical-mechanical generalizations Gentile statistics Haldane–Wu statistics Polychronakos statistics 4 Nonextensive statistics 5 Connection between diﬀerent types of statistics Virial expansion Expansion over statistics parameters 6 Summary

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 2 / 60 Rudolf Clausius — notion of entropy (1865); James Clerk Maxwell — velocity distribution of molecules in an ideal gas (1867); Ludwig Boltzmann — generalization of Maxwell’s results (1871); Josiah Willard Gibbs — paradox related to indistinguishability of particles (1874); Boltzmann — method to calculate the number of microstates (1877); Max Planck — expression S = k log W (1901); Gibbs — ensembles in statistical mechanics and formula for the distribution of probabilities (1902).

Introduction Introduction

Brief history (since the Beginning of Time):

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 3 / 60 James Clerk Maxwell — velocity distribution of molecules in an ideal gas (1867); Ludwig Boltzmann — generalization of Maxwell’s results (1871); Josiah Willard Gibbs — paradox related to indistinguishability of particles (1874); Boltzmann — method to calculate the number of microstates (1877); Max Planck — expression S = k log W (1901); Gibbs — ensembles in statistical mechanics and formula for the distribution of probabilities (1902).

Introduction Introduction

Brief history (since the Beginning of Time): Rudolf Clausius — notion of entropy (1865);

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 3 / 60 Ludwig Boltzmann — generalization of Maxwell’s results (1871); Josiah Willard Gibbs — paradox related to indistinguishability of particles (1874); Boltzmann — method to calculate the number of microstates (1877); Max Planck — expression S = k log W (1901); Gibbs — ensembles in statistical mechanics and formula for the distribution of probabilities (1902).

Introduction Introduction

Brief history (since the Beginning of Time): Rudolf Clausius — notion of entropy (1865); James Clerk Maxwell — velocity distribution of molecules in an ideal gas (1867);

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 3 / 60 Josiah Willard Gibbs — paradox related to indistinguishability of particles (1874); Boltzmann — method to calculate the number of microstates (1877); Max Planck — expression S = k log W (1901); Gibbs — ensembles in statistical mechanics and formula for the distribution of probabilities (1902).

Introduction Introduction

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 3 / 60 Boltzmann — method to calculate the number of microstates (1877); Max Planck — expression S = k log W (1901); Gibbs — ensembles in statistical mechanics and formula for the distribution of probabilities (1902).

Introduction Introduction

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 3 / 60 Max Planck — expression S = k log W (1901); Gibbs — ensembles in statistical mechanics and formula for the distribution of probabilities (1902).

Introduction Introduction

Brief history (since the Beginning of Time): Rudolf Clausius — notion of entropy (1865); James Clerk Maxwell — velocity distribution of molecules in an ideal gas (1867); Ludwig Boltzmann — generalization of Maxwell’s results (1871); Josiah Willard Gibbs — paradox related to indistinguishability of particles (1874); Boltzmann — method to calculate the number of microstates (1877);

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 3 / 60 Gibbs — ensembles in statistical mechanics and formula for the distribution of probabilities (1902).

Introduction Introduction

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 3 / 60 Introduction Introduction

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 3 / 60 Satyendranath Bose (1924) Albert Einstein (1924–25);

Wolfgang Pauli — exclusion principle (1925); Enrico Fermi (1926) Paul Adrien Maurice Dirac (1926);

Goudsmit & Uhlenbeck — notion of spin (1926) Markus Fierz — spin–statistics theorem (1939)

A more systematic approach to this issue was given by Pauli (1940) and Schwinger (1951).

Introduction Introduction

Quantum distributions:

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 4 / 60 Wolfgang Pauli — exclusion principle (1925); Enrico Fermi (1926) Paul Adrien Maurice Dirac (1926);

Goudsmit & Uhlenbeck — notion of spin (1926) Markus Fierz — spin–statistics theorem (1939)

A more systematic approach to this issue was given by Pauli (1940) and Schwinger (1951).

Introduction Introduction

Quantum distributions: Satyendranath Bose (1924) Albert Einstein (1924–25);

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 4 / 60 Wolfgang Pauli — exclusion principle (1925); Enrico Fermi (1926) Paul Adrien Maurice Dirac (1926);

Goudsmit & Uhlenbeck — notion of spin (1926) Markus Fierz — spin–statistics theorem (1939)

A more systematic approach to this issue was given by Pauli (1940) and Schwinger (1951).

Introduction Introduction

Quantum distributions: Satyendranath Bose (1924) Albert Einstein (1924–25);

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 4 / 60 Enrico Fermi (1926) Paul Adrien Maurice Dirac (1926);

Goudsmit & Uhlenbeck — notion of spin (1926) Markus Fierz — spin–statistics theorem (1939)

A more systematic approach to this issue was given by Pauli (1940) and Schwinger (1951).

Introduction Introduction

Quantum distributions: Satyendranath Bose (1924) Albert Einstein (1924–25);

Wolfgang Pauli — exclusion principle (1925);

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 4 / 60 Goudsmit & Uhlenbeck — notion of spin (1926) Markus Fierz — spin–statistics theorem (1939)

A more systematic approach to this issue was given by Pauli (1940) and Schwinger (1951).

Introduction Introduction

Quantum distributions: Satyendranath Bose (1924) Albert Einstein (1924–25);

Wolfgang Pauli — exclusion principle (1925); Enrico Fermi (1926) Paul Adrien Maurice Dirac (1926);

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 4 / 60 Goudsmit & Uhlenbeck — notion of spin (1926) Markus Fierz — spin–statistics theorem (1939)

A more systematic approach to this issue was given by Pauli (1940) and Schwinger (1951).

Introduction Introduction

Quantum distributions: Satyendranath Bose (1924) Albert Einstein (1924–25);

Wolfgang Pauli — exclusion principle (1925); Enrico Fermi (1926) Paul Adrien Maurice Dirac (1926);

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 4 / 60 Markus Fierz — spin–statistics theorem (1939)

A more systematic approach to this issue was given by Pauli (1940) and Schwinger (1951).

Introduction Introduction

Quantum distributions: Satyendranath Bose (1924) Albert Einstein (1924–25);

Wolfgang Pauli — exclusion principle (1925); Enrico Fermi (1926) Paul Adrien Maurice Dirac (1926);

Goudsmit & Uhlenbeck — notion of spin (1926)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 4 / 60 A more systematic approach to this issue was given by Pauli (1940) and Schwinger (1951).

Introduction Introduction

Quantum distributions: Satyendranath Bose (1924) Albert Einstein (1924–25);

Wolfgang Pauli — exclusion principle (1925); Enrico Fermi (1926) Paul Adrien Maurice Dirac (1926);

Goudsmit & Uhlenbeck — notion of spin (1926) Markus Fierz — spin–statistics theorem (1939)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 4 / 60 A more systematic approach to this issue was given by Pauli (1940) and Schwinger (1951).

Introduction Introduction

Quantum distributions: Satyendranath Bose (1924) Albert Einstein (1924–25);

Wolfgang Pauli — exclusion principle (1925); Enrico Fermi (1926) Paul Adrien Maurice Dirac (1926);

Goudsmit & Uhlenbeck — notion of spin (1926) Markus Fierz — spin–statistics theorem (1939)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 4 / 60 Introduction Introduction

Quantum distributions: Satyendranath Bose (1924) Albert Einstein (1924–25);

Wolfgang Pauli — exclusion principle (1925); Enrico Fermi (1926) Paul Adrien Maurice Dirac (1926);

Goudsmit & Uhlenbeck — notion of spin (1926) Markus Fierz — spin–statistics theorem (1939)

A more systematic approach to this issue was given by Pauli (1940) and Schwinger (1951). A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 4 / 60 Such a division is quite conditional as both approaches might be linked tightly sometimes.

Introduction Introduction

To put the things simple, two approaches to generalize quantum distributions can be distinguished: quantum-mechanical considerations, namely, analysis of wave function properties, commutators, etc. methods from the statistical physics, namely, counting of microstates, generalization of entropy, etc.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 5 / 60 Introduction Introduction

Such a division is quite conditional as both approaches might be linked tightly sometimes.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 5 / 60 Herbert Sydney Green (1953) — parastatistics: parabosons are anti-symmetric wrt permutation of no more than k particles while parafermions are symmetric in such a case. q-deformed commutators or q-mutators † † † [a, a ]q = aa − qa a. (Oscar Wallace Greenberg, 1990)

Pubblicazioni scientifiche: “Le statistiche intermedie e le proprietà dell’elio liquido”, Rendiconti del Seminario Matematico e fisico di Milano XV (1941-XIX), 1-19, 1941.

Introduction

1940 Introduction Pubblicazioni scientifiche: “Sulle equazioni “Sopra il fenomeno della Condensazione del gas d’onda relativistiche di Dirac per particelle con di Bose-Einstein”, La Ricerca scientifica, Anno momento intrinseco qualsiasi”, Il Nuovo 12°. N. 3, 3-8, marzo 1941. Cimento, a. XVII, n. 1, 5-12, gennaio 1940. “Osservazioni sopra le statistiche intermedie”, Il Nuovo Cimento, a. XVII, n. 10, 1-5, dicembre 1942 1940. “Per la teoria del modello vettoriale Muore a Milano il 30 marzo. Statistics generalizations: dell’atomo”, Rendiconti R. Istituto Lombardo, Classe di Scienze, vol. LXXIV, fasc. I, 1-7, 1940-41. “Sopra una supposta non validità del principio galileiano della composizione dei moti Giovanni Gentile (jr.) (1940) nella fisica atomica”, Rendiconti R. Istituto Lombardo, Classe di Scienze, vol. LXXIV, fasc. I, 1-4, 1940-41. “Osservazioni sopra le statistiche intermedie”, Rendiconti R. Istituto Lombardo, — intermediate statistics, with maximalClasse di Scienze, vol. LXXIV, fasc. I, 33-37, occupation of a level limited by some1940-41. ﬁnite s. 1941 Viene nominato professore ordinario della cattedra di Fisica teorica presso l’Università degli studi di Milano. Riceve la nomina di Socio corrispondente del Reale Istituto Lombardo di Scienze e Lettere, nella Classe di Scienze matematiche e naturali, sezione di scienze fisiche, chimiche.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 6 / 60 q-deformed commutators or q-mutators † † † [a, a ]q = aa − qa a. (Oscar Wallace Greenberg, 1990)

Pubblicazioni scientifiche: “Le statistiche intermedie e le proprietà dell’elio liquido”, Rendiconti del Seminario Matematico e fisico di Milano XV (1941-XIX), 1-19, 1941.

Introduction

1940 Introduction Pubblicazioni scientifiche: “Sulle equazioni “Sopra il fenomeno della Condensazione del gas d’onda relativistiche di Dirac per particelle con di Bose-Einstein”, La Ricerca scientifica, Anno momento intrinseco qualsiasi”, Il Nuovo 12°. N. 3, 3-8, marzo 1941. Cimento, a. XVII, n. 1, 5-12, gennaio 1940. “Osservazioni sopra le statistiche intermedie”, Il Nuovo Cimento, a. XVII, n. 10, 1-5, dicembre 1942 1940. “Per la teoria del modello vettoriale Muore a Milano il 30 marzo. Statistics generalizations: dell’atomo”, Rendiconti R. Istituto Lombardo, Classe di Scienze, vol. LXXIV, fasc. I, 1-7, 1940-41. “Sopra una supposta non validità del principio galileiano della composizione dei moti Giovanni Gentile (jr.) (1940) nella fisica atomica”, Rendiconti R. Istituto Lombardo, Classe di Scienze, vol. LXXIV, fasc. I, 1-4, 1940-41. “Osservazioni sopra le statistiche intermedie”, Rendiconti R. Istituto Lombardo, — intermediate statistics, with maximalClasse di Scienze, vol. LXXIV, fasc. I, 33-37, occupation of a level limited by some1940-41. ﬁnite s. 1941 Viene nominato professore ordinario della cattedra di Fisica teorica presso l’Università degli studi di Milano. Riceve la nomina di Socio corrispondente del Herbert Sydney Green (1953) —Reale parastatistics: Istituto Lombardo di Scienze e Lettere, nella Classe di Scienze matematiche e naturali, parabosons are anti-symmetric wrtsezione di permutation scienze fisiche, chimiche. of no more than k particles while parafermions 4 are symmetric in such a case.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 6 / 60 Pubblicazioni scientifiche: “Le statistiche intermedie e le proprietà dell’elio liquido”, Rendiconti del Seminario Matematico e fisico di Milano XV (1941-XIX), 1-19, 1941.

Introduction

1940 Introduction Pubblicazioni scientifiche: “Sulle equazioni “Sopra il fenomeno della Condensazione del gas d’onda relativistiche di Dirac per particelle con di Bose-Einstein”, La Ricerca scientifica, Anno momento intrinseco qualsiasi”, Il Nuovo 12°. N. 3, 3-8, marzo 1941. Cimento, a. XVII, n. 1, 5-12, gennaio 1940. “Osservazioni sopra le statistiche intermedie”, Il Nuovo Cimento, a. XVII, n. 10, 1-5, dicembre 1942 1940. “Per la teoria del modello vettoriale Muore a Milano il 30 marzo. Statistics generalizations: dell’atomo”, Rendiconti R. Istituto Lombardo, Classe di Scienze, vol. LXXIV, fasc. I, 1-7, 1940-41. “Sopra una supposta non validità del principio galileiano della composizione dei moti Giovanni Gentile (jr.) (1940) nella fisica atomica”, Rendiconti R. Istituto Lombardo, Classe di Scienze, vol. LXXIV, fasc. I, 1-4, 1940-41. “Osservazioni sopra le statistiche intermedie”, Rendiconti R. Istituto Lombardo, — intermediate statistics, with maximalClasse di Scienze, vol. LXXIV, fasc. I, 33-37, occupation of a level limited by some1940-41. ﬁnite s. 1941 Viene nominato professore ordinario della cattedra di Fisica teorica presso l’Università degli studi di Milano. Riceve la nomina di Socio corrispondente del Herbert Sydney Green (1953) —Reale parastatistics: Istituto Lombardo di Scienze e Lettere, nella Classe di Scienze matematiche e naturali, parabosons are anti-symmetric wrtsezione di permutation scienze fisiche, chimiche. of no more than k particles while parafermions 4 are symmetric in such a case. q-deformed commutators or q-mutators † † † [a, a ]q = aa − qa a. (Oscar Wallace Greenberg, 1990)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 6 / 60 Frank Wilczek (1982) — term anyon (from English ‘any’).

Introduction Introduction

Jon Manne Leinaas & Jan Myrheim (1977) showed that in a 2D system the wavefunction phase can change arbitrarily at the permutation of two particles.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 7 / 60 Introduction Introduction

Jon Manne Leinaas & Jan Myrheim (1977) showed that in a 2D system the wavefunction phase can change arbitrarily at the permutation of two particles.

Frank Wilczek (1982) — term anyon (from English ‘any’).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 7 / 60 Yong-Shi Wu (1994) — distribution function in the Haldane statistics (so called fractional exclusion statistics).

Constantino Tsallis (1988) — nonextensive statistics (systems with nonadditive entropy, i.e. with long-range interactions, ‘memory’ eﬀects, strongly non-Markovian processes, etc.).

Introduction Introduction

Statistics generalizations: F. Duncan M. Haldane (1991) — generalization of the Pauli exclusion principle; interpolation between the Bose and Fermi limits for the number of microstates.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 8 / 60 Constantino Tsallis (1988) — nonextensive statistics (systems with nonadditive entropy, i.e. with long-range interactions, ‘memory’ eﬀects, strongly non-Markovian processes, etc.).

Introduction Introduction

Statistics generalizations: F. Duncan M. Haldane (1991) — generalization of the Pauli exclusion principle; interpolation between the Bose and Fermi limits for the number of microstates. Yong-Shi Wu (1994) — distribution function in the Haldane statistics (so called fractional exclusion statistics).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 8 / 60 Introduction Introduction

Constantino Tsallis (1988) — nonextensive statistics (systems with nonadditive entropy, i.e. with long-range interactions, ‘memory’ eﬀects, strongly non-Markovian processes, etc.).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 8 / 60 Permutation operator P12:

iπα P12ψ(1, 2) = ψ(2, 1) = e ψ(1, 2). (1)

Repeating its action leads to:

2 2iπα P12ψ(1, 2) = e ψ(1, 2) = ψ(1, 2), (2)

so that α = 0 or 1, corresponding to the symmetric (bosons) or anti-symmetric (fermions) wavefunction. This is linked to the fact that the double permutation is the identity 2 (unit) operation, P12 = I . However, this appears to be a property of the 3D space, not in lower dimensions.

Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 9 / 60 Repeating its action leads to:

2 2iπα P12ψ(1, 2) = e ψ(1, 2) = ψ(1, 2), (2)

Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

Permutation operator P12:

iπα P12ψ(1, 2) = ψ(2, 1) = e ψ(1, 2). (1)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 9 / 60 This is linked to the fact that the double permutation is the identity 2 (unit) operation, P12 = I . However, this appears to be a property of the 3D space, not in lower dimensions.

Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

Permutation operator P12:

iπα P12ψ(1, 2) = ψ(2, 1) = e ψ(1, 2). (1)

Repeating its action leads to:

2 2iπα P12ψ(1, 2) = e ψ(1, 2) = ψ(1, 2), (2) so that α = 0 or 1, corresponding to the symmetric (bosons) or anti-symmetric (fermions) wavefunction.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 9 / 60 Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

Permutation operator P12:

iπα P12ψ(1, 2) = ψ(2, 1) = e ψ(1, 2). (1)

Repeating its action leads to:

2 2iπα P12ψ(1, 2) = e ψ(1, 2) = ψ(1, 2), (2) so that α = 0 or 1, corresponding to the symmetric (bosons) or anti-symmetric (fermions) wavefunction. This is linked to the fact that the double permutation is the identity 2 (unit) operation, P12 = I . However, this appears to be a property of the 3D space, not in lower dimensions.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 9 / 60 2 So, in the 2D space double permutation P12 6= I and there is no restrictions for the wavefunction phase.

Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

The permutation operation can be [clasically] thought as a motion of one particle around another. In the 3D space such a closed path can be shrunk to a point, but this is not the case in the 2D space due to the presence of a hard-core:

Figure: Double permutation in 2D (left) is not identity as in 3D (right).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 10 / 60 Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

Figure: Double permutation in 2D (left) is not identity as in 3D (right).

2 So, in the 2D space double permutation P12 6= I and there is no restrictions for the wavefunction phase.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 10 / 60 Its generators σi are:

Figure: Action of the operator σ1. The braid connecting lower point 1 with upper point 2 goes over the other braid. Such an operator corresponds to the permutation of particles 1 and 2 (in a deﬁned direction, say, counterclockwise).

Quantum-mechanical approach Anyons Braid group

In the 3D space — permutation group SN .

In the 2D space — so called braid group BN .

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 11 / 60 Such an operator corresponds to the permutation of particles 1 and 2 (in a deﬁned direction, say, counterclockwise).

Quantum-mechanical approach Anyons Braid group

In the 3D space — permutation group SN .

In the 2D space — so called braid group BN .

Its generators σi are:

Figure: Action of the operator σ1. The braid connecting lower point 1 with upper point 2 goes over the other braid.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 11 / 60 Quantum-mechanical approach Anyons Braid group

In the 3D space — permutation group SN .

In the 2D space — so called braid group BN .

Its generators σi are:

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 11 / 60 Figure: Double permutation is not the identity operation: 2 −1 σ1 6= σ1 σ1 = I .

Quantum-mechanical approach Anyons Braid group

Repeating the action of σ1 does not lead to the initial conﬁguration: 2 −1 the braid is “plaited”, i.e. topologically σ1 6= σ1 σ1 = I .

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 12 / 60 Figure: Double permutation is not the identity operation: 2 −1 σ1 6= σ1 σ1 = I .

Quantum-mechanical approach Anyons Braid group

Repeating the action of σ1 does not lead to the initial conﬁguration: 2 −1 the braid is “plaited”, i.e. topologically σ1 6= σ1 σ1 = I .

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 12 / 60 Quantum-mechanical approach Anyons Braid group

Repeating the action of σ1 does not lead to the initial conﬁguration: 2 −1 the braid is “plaited”, i.e. topologically σ1 6= σ1 σ1 = I .

Figure: Double permutation is not the identity operation: 2 −1 σ1 6= σ1 σ1 = I .

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 12 / 60 Figure: Graphical interpretation of the property σi σi+1σi = σi+1σi σi+1.

Quantum-mechanical approach Anyons Braid group

Generations of the braid group σi satisfy the following relations (Artin relations):

σi σi+1σi = σi+1σi σi+1, (3)

σi σj = σj σi , if |i − j| ≥ 2. (4)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 13 / 60 Quantum-mechanical approach Anyons Braid group

Generations of the braid group σi satisfy the following relations (Artin relations):

σi σi+1σi = σi+1σi σi+1, (3)

σi σj = σj σi , if |i − j| ≥ 2. (4)

Figure: Graphical interpretation of the property σi σi+1σi = σi+1σi σi+1.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 13 / 60 Quantum-mechanical approach Anyons Braid group

Figure: Graphical interpretation of the property σi σj = σj σi , |i − j| ≥ 2.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 14 / 60 Camino et al. (2005) reported about the observation of the interference pattern corresponding to the Laughlin quasi-particles (elementary excitations with a fractional charge characteristic to the fractional quantum Hall eﬀect; they are anyon candidates),.

Experimental setup to observe anyons in a system consisting of a superconducting ﬁlm on a semiconductor heterotransition was suggested by Weeks et al. (2007) and another one involving one-dimensional optical lattices was proposed by Keilmann et al. (2011).

Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

The ﬁrst physical model of anyons as a composite of a magnetic ﬂux tube and a charged particle was suggested by Wilczek (1982).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 15 / 60 Experimental setup to observe anyons in a system consisting of a superconducting ﬁlm on a semiconductor heterotransition was suggested by Weeks et al. (2007) and another one involving one-dimensional optical lattices was proposed by Keilmann et al. (2011).

Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

The ﬁrst physical model of anyons as a composite of a magnetic ﬂux tube and a charged particle was suggested by Wilczek (1982).

Camino et al. (2005) reported about the observation of the interference pattern corresponding to the Laughlin quasi-particles (elementary excitations with a fractional charge characteristic to the fractional quantum Hall eﬀect; they are anyon candidates),.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 15 / 60 Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

The ﬁrst physical model of anyons as a composite of a magnetic ﬂux tube and a charged particle was suggested by Wilczek (1982).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 15 / 60 The fractional quantum Hall eﬀect (FQHE) means a precise quantization of the Hall conductance in 2D electron systems at fractional values of e2/h.

Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

The 1998 Nobel Prize in Physics was awarded jointly to Robert Laughlin, Horst St¨ormer,and Daniel Tsui for their discovery of a new form of quantum ﬂuid with fractionally charged excitations.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 16 / 60 Quantum-mechanical approach Anyons Quantum-mechanical approach: Anyons

The 1998 Nobel Prize in Physics was awarded jointly to Robert Laughlin, Horst St¨ormer,and Daniel Tsui for their discovery of a new form of quantum ﬂuid with fractionally charged excitations.

The fractional quantum Hall eﬀect (FQHE) means a precise quantization of the Hall conductance in 2D electron systems at fractional values of e2/h.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 16 / 60 Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

Commutation relations for the creation–annihilation operators a†, a can be interpolated between fermions (anticommutator) and bosons (ordinary commutator) introducing q-deformed commutators or q-mutators. The simplest generalization is the so-called quon algebra (Greenberg 1990; Mohapatra 1990; Greenberg 1991):

† † † [aj , ak ]q = aj ak − qak aj = δjk , (5) where −1 ≤ q ≤ 1 provides a continuous interpolation between q = −1 (fermions) and q = 1 (bosons).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 17 / 60 Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

Introducing the operator for the “number of particles” N, supplementing the q-mutator by the respective relations:

aa† − qa†a = 1, [N, a] = −a, [N, a†] = a†, (6) one obtains a modiﬁed algebra for new operators

c = q−λN/2a, c† = a†q−λN/2, (7) where λ is a real number. In this new algebra the commutation relation is:

cc† − q1−λc†c = q−λN , (8) and usually one takes λ = 1 or λ = 1/2.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 18 / 60 Orthonormalized set of eigenstates is given by:

1 † n |ni = p (a ) |0i, a|0i = 0, (11) [n]q! where the q-factorial

[n]q! = [n]q[n − 1]q ... [1]q (12) = (qn−1 + ... q + 1)(qn−2 + ... q + 1) ... (q + 1)1;

[0]q! = 1 is deﬁned via so-called q-numbers: qn − 1 [n] = , [n] = n. (13) q q − 1 1

Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

Algebra of q-bosonic operators: aa† − qa†a = 1, [a, a] = [a†, a†] = 0, (9) [N, a†] = a†, [N, a] = −a. (10)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 19 / 60 where the q-factorial

[n]q! = [n]q[n − 1]q ... [1]q (12) = (qn−1 + ... q + 1)(qn−2 + ... q + 1) ... (q + 1)1;

[0]q! = 1 is deﬁned via so-called q-numbers: qn − 1 [n] = , [n] = n. (13) q q − 1 1

Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

Algebra of q-bosonic operators: aa† − qa†a = 1, [a, a] = [a†, a†] = 0, (9) [N, a†] = a†, [N, a] = −a. (10) Orthonormalized set of eigenstates is given by:

1 † n |ni = p (a ) |0i, a|0i = 0, (11) [n]q!

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 19 / 60 Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

Algebra of q-bosonic operators: aa† − qa†a = 1, [a, a] = [a†, a†] = 0, (9) [N, a†] = a†, [N, a] = −a. (10) Orthonormalized set of eigenstates is given by:

1 † n |ni = p (a ) |0i, a|0i = 0, (11) [n]q! where the q-factorial

[n]q! = [n]q[n − 1]q ... [1]q (12) = (qn−1 + ... q + 1)(qn−2 + ... q + 1) ... (q + 1)1;

[0]q! = 1 is deﬁned via so-called q-numbers: qn − 1 [n] = , [n] = n. (13) q q − 1 1 A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 19 / 60 It can be shown that in the grand canonical ensemble occupation numbers for the ith level with energy εi are:

1 z −1eβεi − 1 ni = ln , (18) ln q z −1eβεi − q where z = eβµ is fugacity, µ is chemical potential, and β = 1/T .

Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

These operators act on eigenstates as follows:

† q a |ni = [n + 1]q|n + 1i, (14) q a|ni = [n]q|n − 1i, (15) N|ni = n|ni, (16) while a†a = [N], aa† = [N + 1]. (17)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 20 / 60 Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

These operators act on eigenstates as follows:

† q a |ni = [n + 1]q|n + 1i, (14) q a|ni = [n]q|n − 1i, (15) N|ni = n|ni, (16) while a†a = [N], aa† = [N + 1]. (17)

It can be shown that in the grand canonical ensemble occupation numbers for the ith level with energy εi are:

1 z −1eβεi − 1 ni = ln , (18) ln q z −1eβεi − q where z = eβµ is fugacity, µ is chemical potential, and β = 1/T . A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 20 / 60 Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

R.-Monteiro et al. (1993) considered the following q-bosonic algebra:

aa† − qa†a = q−N , [N, a†] = a†, [N, a] = −a, (19) and for the Hamiltonian

† † † N/2 N/2 H = ~ωA A, A = a q , A = q a, (20) in the limit of q → ∞ obtained a fermion-like expression for occupation numbers:

2 1 + 2e−~ωβq hNi ' . (21) 1 + e~ωβ + e−~ωβq2

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 21 / 60 Dutt et al. (1994) considered relations between the creation and annihilation operators of q-fermions: 1 1 † 2 † − 2 Nf fqfq + q fq fq = q , (22) † † [Nf , fq] = −fq, [Nf , fq ] = fq . (23) Yang et al. (1998) suggested the following variant: † † † ∗ † † † aqaq − qaqaq = 1, aq∗ aq∗ − q aq∗ aq∗ = 1. aq = (aq∗ ) , (24) where q = e2πi/(s+1) is the (s + 1)th root of unity. As with fq, these operators have the nilpotence property: n n † n † n (aq) = (aq∗ ) = aq = (aq∗ ) = 0, if n ≥ s + 1. (25) So, such an algebra corresponds to limiting the occupation of a quantum state by some s.

Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

With a complex parameter q on the unit circle |q| = 1,

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 22 / 60 Yang et al. (1998) suggested the following variant: † † † ∗ † † † aqaq − qaqaq = 1, aq∗ aq∗ − q aq∗ aq∗ = 1. aq = (aq∗ ) , (24) where q = e2πi/(s+1) is the (s + 1)th root of unity. As with fq, these operators have the nilpotence property: n n † n † n (aq) = (aq∗ ) = aq = (aq∗ ) = 0, if n ≥ s + 1. (25) So, such an algebra corresponds to limiting the occupation of a quantum state by some s.

Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 22 / 60 As with fq, these operators have the nilpotence property: n n † n † n (aq) = (aq∗ ) = aq = (aq∗ ) = 0, if n ≥ s + 1. (25) So, such an algebra corresponds to limiting the occupation of a quantum state by some s.

Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 22 / 60 So, such an algebra corresponds to limiting the occupation of a quantum state by some s.

Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

With a complex parameter q on the unit circle |q| = 1, Dutt et al. (1994) considered relations between the creation and annihilation operators of q-fermions: 1 1 † 2 † − 2 Nf fqfq + q fq fq = q , (22) † † [Nf , fq] = −fq, [Nf , fq ] = fq . (23) Yang et al. (1998) suggested the following variant: † † † ∗ † † † aqaq − qaqaq = 1, aq∗ aq∗ − q aq∗ aq∗ = 1. aq = (aq∗ ) , (24) where q = e2πi/(s+1) is the (s + 1)th root of unity. As with fq, these operators have the nilpotence property: n n † n † n (aq) = (aq∗ ) = aq = (aq∗ ) = 0, if n ≥ s + 1. (25)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 22 / 60 Quantum-mechanical approach q-deformations Quantum-mechanical approach: q-deformations

[[a†, a] , a†] = 2a† — parafermions. . .

Quantum-mechanical approach Parastatistics Quantum-mechanical approach: parastatistics

Just in a glance:

[a, a†] = 1 — bosons; {a, a†} = 1 — fermions;

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 23 / 60 [[a†, a] , a†] = 2a† — parafermions. . .

Quantum-mechanical approach Parastatistics Quantum-mechanical approach: parastatistics

Just in a glance:

[a, a†] = 1 — bosons; {a, a†} = 1 — fermions;

[{a†, a}, a†] = 2a† — parabosons;

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 23 / 60 Quantum-mechanical approach Parastatistics Quantum-mechanical approach: parastatistics

Just in a glance:

[a, a†] = 1 — bosons; {a, a†} = 1 — fermions;

[{a†, a}, a†] = 2a† — parabosons;

[[a†, a] , a†] = 2a† — parafermions. . .

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 23 / 60 To ﬁnd the occupation number ni = Ni /Gi (distribution function) corresponding to a certain type of statistics, the following method might be applied. Entropy S of the system is linked to the number of microstates W via: Y S = ln W , where W = Wi (Ni ). (27) i

Statistical-mechanical generalizations Statistical-mechanical generalizations

Fractional (intermediate) statistics can be obtained by means of interpolation between expressions for the number of microstates Wi in bosonic (B) and fermionic (F) limits:

B (Gi + Ni − 1)! F Gi ! Wi = , Wi = , (26) Ni !(Gi − 1)! Ni !(Gi − Ni )! where Gi is the degeneration of the ith level, and Ni is the number of particles on this level.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 24 / 60 Statistical-mechanical generalizations Statistical-mechanical generalizations

Fractional (intermediate) statistics can be obtained by means of interpolation between expressions for the number of microstates Wi in bosonic (B) and fermionic (F) limits:

B (Gi + Ni − 1)! F Gi ! Wi = , Wi = , (26) Ni !(Gi − 1)! Ni !(Gi − Ni )! where Gi is the degeneration of the ith level, and Ni is the number of particles on this level.

To ﬁnd the occupation number ni = Ni /Gi (distribution function) corresponding to a certain type of statistics, the following method might be applied. Entropy S of the system is linked to the number of microstates W via: Y S = ln W , where W = Wi (Ni ). (27) i

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 24 / 60 Expression for ni can be obtained by ﬁnding an extremum of this functional at additional constraints ﬁxing the number of particles in the system X N = Ni , (28) i and the total energy X E = εi Ni , (29) i

where εi is the energy of the ith level. Thus, we face the conditional extremum problem: δS + αδN − βδE = 0, (30)

where variations are wrt Ni and Lagrange multipliers are linked to temperature T and chemical potential µ: α = µ/T , β = 1/T .

Statistical-mechanical generalizations Statistical-mechanical generalizations

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 25 / 60 Thus, we face the conditional extremum problem: δS + αδN − βδE = 0, (30)

where variations are wrt Ni and Lagrange multipliers are linked to temperature T and chemical potential µ: α = µ/T , β = 1/T .

Statistical-mechanical generalizations Statistical-mechanical generalizations

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 25 / 60 Statistical-mechanical generalizations Statistical-mechanical generalizations

Expression for ni can be obtained by ﬁnding an extremum of this functional at additional constraints ﬁxing the number of particles in the system X N = Ni , (28) i and the total energy X E = εi Ni , (29) i where εi is the energy of the ith level. Thus, we face the conditional extremum problem: δS + αδN − βδE = 0, (30) where variations are wrt Ni and Lagrange multipliers are linked to temperature T and chemical potential µ: α = µ/T , β = 1/T . A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 25 / 60 Statistical-mechanical generalizations Gentile statistics Gentile statistics

Let us postulate an intermediate distribution, in which the maximal occupation of a level is limited by a ﬁnite number s. Such a statistics is know as the Gentile statistics (Gentile 1940). It is clear that the s = 1 limit corresponds to the Fermi distribution, and the s = ∞ describes the Bose distribution. Expression for occupation numbers in the Gentile statistics can be obtained using the expression for the number of possibilities to distribute particles over all possible energy levels in the form:

Y Gi ! W = , ni (0)!ni (1)! ... ni (s)! i where s X Gi = ni (j) (31) j=0 is the weighting coeﬃcient of the ith state. A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 26 / 60 We thus have variational problem (30) with the number of particles and energy given by (28) and (29), respectively. Solving this problem, one derives the occupation numbers:

G 1 s + 1 ni = − , (33) z −1eεi /T − 1 z −(s+1)e(s+1)εi /T − 1

where z = eµ/T is fugacity. After a simple exercise we do obtain the Fermi distribution at s = 1 and the Bose distribution at s = ∞.

Statistical-mechanical generalizations Gentile statistics Gentile statistics

The number of particles as the ith level is

s X Ni = jni (j). (32) j=0

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 27 / 60 Statistical-mechanical generalizations Gentile statistics Gentile statistics

The number of particles as the ith level is

s X Ni = jni (j). (32) j=0

We thus have variational problem (30) with the number of particles and energy given by (28) and (29), respectively. Solving this problem, one derives the occupation numbers:

G 1 s + 1 ni = − , (33) z −1eεi /T − 1 z −(s+1)e(s+1)εi /T − 1 where z = eµ/T is fugacity. After a simple exercise we do obtain the Fermi distribution at s = 1 and the Bose distribution at s = ∞.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 27 / 60 in the low-temperature limit similarities with the Fermi-statistics are evident, in particular, there exists an analog of the Fermi level εG. Equation of state of an ideal D-dimensional gas of N particles with d the dispersion law εp = ap obeying the Gentile statistics is given by: ∞ j−1 pV X N G = D/d bj , (34) NT AD,d VT j=1 G where AD,d is a constant, and virial coeﬃcients bj are linked to that B of the Bose-system bj : G B bj = bj , if j ≤ s, s bG = bB + . (35) s+1 s+1 (s + 1)D/d

Statistical-mechanical generalizations Gentile statistics Gentile statistics

High-temperature behavior is similar to that in the Bose-statistics;

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 28 / 60 Equation of state of an ideal D-dimensional gas of N particles with d the dispersion law εp = ap obeying the Gentile statistics is given by: ∞ j−1 pV X N G = D/d bj , (34) NT AD,d VT j=1 G where AD,d is a constant, and virial coeﬃcients bj are linked to that B of the Bose-system bj : G B bj = bj , if j ≤ s, s bG = bB + . (35) s+1 s+1 (s + 1)D/d

Statistical-mechanical generalizations Gentile statistics Gentile statistics

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 28 / 60 Statistical-mechanical generalizations Gentile statistics Gentile statistics

High-temperature behavior is similar to that in the Bose-statistics; in the low-temperature limit similarities with the Fermi-statistics are evident, in particular, there exists an analog of the Fermi level εG. Equation of state of an ideal D-dimensional gas of N particles with d the dispersion law εp = ap obeying the Gentile statistics is given by: ∞ j−1 pV X N G = D/d bj , (34) NT AD,d VT j=1 G where AD,d is a constant, and virial coeﬃcients bj are linked to that B of the Bose-system bj : G B bj = bj , if j ≤ s, s bG = bB + . (35) s+1 s+1 (s + 1)D/d

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 28 / 60 Gentile statistics was used to solve the problem of restricted partitions in number theory (Srivatsan et al. 2006).

One can show that with a special choice of the statistics parameter s a ﬁnite Bose-system can be modeled (Rovenchak 2009).

A certain (incomplete) equivalence of the Gentile statistics and the anyonic statistics was established by Shen et al. (2010).

Statistical-mechanical generalizations Gentile statistics Gentile statistics

In the high-temperature limit, the heat capacity of such a system CV B is also linked to that of the Bose-gas CV :

B s CV CV 1 Ds Ds N = − D/d − 1 D/d + .... N N (s + 1) d d AD,d VT

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 29 / 60 One can show that with a special choice of the statistics parameter s a ﬁnite Bose-system can be modeled (Rovenchak 2009).

A certain (incomplete) equivalence of the Gentile statistics and the anyonic statistics was established by Shen et al. (2010).

Statistical-mechanical generalizations Gentile statistics Gentile statistics

In the high-temperature limit, the heat capacity of such a system CV B is also linked to that of the Bose-gas CV :

B s CV CV 1 Ds Ds N = − D/d − 1 D/d + .... N N (s + 1) d d AD,d VT

Gentile statistics was used to solve the problem of restricted partitions in number theory (Srivatsan et al. 2006).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 29 / 60 A certain (incomplete) equivalence of the Gentile statistics and the anyonic statistics was established by Shen et al. (2010).

Statistical-mechanical generalizations Gentile statistics Gentile statistics

In the high-temperature limit, the heat capacity of such a system CV B is also linked to that of the Bose-gas CV :

B s CV CV 1 Ds Ds N = − D/d − 1 D/d + .... N N (s + 1) d d AD,d VT

Gentile statistics was used to solve the problem of restricted partitions in number theory (Srivatsan et al. 2006).

One can show that with a special choice of the statistics parameter s a ﬁnite Bose-system can be modeled (Rovenchak 2009).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 29 / 60 Statistical-mechanical generalizations Gentile statistics Gentile statistics

In the high-temperature limit, the heat capacity of such a system CV B is also linked to that of the Bose-gas CV :

B s CV CV 1 Ds Ds N = − D/d − 1 D/d + .... N N (s + 1) d d AD,d VT

Gentile statistics was used to solve the problem of restricted partitions in number theory (Srivatsan et al. 2006).

One can show that with a special choice of the statistics parameter s a ﬁnite Bose-system can be modeled (Rovenchak 2009).

A certain (incomplete) equivalence of the Gentile statistics and the anyonic statistics was established by Shen et al. (2010).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 29 / 60 It is clear that g = 1 corresponds to fermions (adding one particle removes one state according the the Pauli principle) and g = 0 corresponds to bosons (no limitations for state occupation). In fact, Haldane’s proposal consists in postulating a certain generalized Pauli principle concerning several states.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

Haldane (1991) proposed to introduce the parameter of statistical interaction d − d g = − N+∆N N , (36) ∆N where dN is the dimensionality of the space of single-particle states for the system of N particles provided that coordinates of the remaining N − 1 particles are ﬁxed.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 30 / 60 In fact, Haldane’s proposal consists in postulating a certain generalized Pauli principle concerning several states.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

It is clear that g = 1 corresponds to fermions (adding one particle removes one state according the the Pauli principle) and g = 0 corresponds to bosons (no limitations for state occupation).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 30 / 60 Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

It is clear that g = 1 corresponds to fermions (adding one particle removes one state according the the Pauli principle) and g = 0 corresponds to bosons (no limitations for state occupation). In fact, Haldane’s proposal consists in postulating a certain generalized Pauli principle concerning several states.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 30 / 60 Mean occupation numbers ni = Ni /Gi are given by: 1 ni = , (38) w (e(εi −µ)/T ) + g where the function w(x) solves such a transcendental equation w g (x) [1 + w(x)]1−g = x ≡ e(εi −µ)/T . (39) At g = 0 one obtains w(x) = x − 1 (Bose distribution), and at g = 1 the function w(x) = x leading to the Fermi distribution.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

The number of microstates Wi can be interpolated using the following formula (Wu 1994):

[Gi + (Ni − 1)(1 − g)]! Wi = (37) Ni ![Gi − gNi − (1 − g)]! B F reducing to Wi in the limit of g = 0 and to Wi as g = 1, respectively.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 31 / 60 At g = 0 one obtains w(x) = x − 1 (Bose distribution), and at g = 1 the function w(x) = x leading to the Fermi distribution.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

The number of microstates Wi can be interpolated using the following formula (Wu 1994):

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 31 / 60 Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

The number of microstates Wi can be interpolated using the following formula (Wu 1994):

[Gi + (Ni − 1)(1 − g)]! Wi = (37) Ni ![Gi − gNi − (1 − g)]! B F reducing to Wi in the limit of g = 0 and to Wi as g = 1, respectively. Mean occupation numbers ni = Ni /Gi are given by: 1 ni = , (38) w (e(εi −µ)/T ) + g where the function w(x) solves such a transcendental equation w g (x) [1 + w(x)]1−g = x ≡ e(εi −µ)/T . (39) At g = 0 one obtains w(x) = x − 1 (Bose distribution), and at g = 1 the function w(x) = x leading to the Fermi distribution. A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 31 / 60 Eq. (39) can be solved analytically for some values of g beyond the 1 1 1 2 3 3 4 Bose and Fermi limits, namely: g = 2 , 2, 3 , 3, 4 , 4, 3 , 2 , 4 , 3 . The simplest result is obtained for g = 1/2 corresponding to so called semions: 1 ni = p . (41) 1/4 + e2(εi −µ)/T Generally, for particles obeying the Haldane–Wu statistics the terms ‘excluson’) or ‘g-on’ are used.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 32 / 60 Generally, for particles obeying the Haldane–Wu statistics the terms ‘excluson’) or ‘g-on’ are used.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 32 / 60 Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

It is quite straightforward to show that at T = 0 the behavior of occupation numbers resembles the Fermi statistics: 1/g, if εi < µ0, ni = (40) 0, if εi > µ0, where µ0 is an analog of the Fermi energy. Eq. (39) can be solved analytically for some values of g beyond the 1 1 1 2 3 3 4 Bose and Fermi limits, namely: g = 2 , 2, 3 , 3, 4 , 4, 3 , 2 , 4 , 3 . The simplest result is obtained for g = 1/2 corresponding to so called semions: 1 ni = p . (41) 1/4 + e2(εi −µ)/T Generally, for particles obeying the Haldane–Wu statistics the terms ‘excluson’) or ‘g-on’ are used. A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 32 / 60 This statistics also can be applied to describe 2D electron gas with short-range interactions (Bhaduri et al. 1996) Three-particle Calogero model with −1/4 < λ < 0 can emulate the anyonic statistics (Sree Ranjani et al. 2009).

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

It appears that interacting fermions in the Calogero–Sutherland model with the Hamiltonian (~ = m = 1)

N 2 X 1 ∂ 1 2 2 X λ g(g − 1) H = − 2 + ω xi + 2 , λ = , 2 ∂x 2 (xi − xj ) 2 i=1 i 1≤i

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 33 / 60 Three-particle Calogero model with −1/4 < λ < 0 can emulate the anyonic statistics (Sree Ranjani et al. 2009).

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

It appears that interacting fermions in the Calogero–Sutherland model with the Hamiltonian (~ = m = 1)

N 2 X 1 ∂ 1 2 2 X λ g(g − 1) H = − 2 + ω xi + 2 , λ = , 2 ∂x 2 (xi − xj ) 2 i=1 i 1≤i

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 33 / 60 Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

It appears that interacting fermions in the Calogero–Sutherland model with the Hamiltonian (~ = m = 1)

N 2 X 1 ∂ 1 2 2 X λ g(g − 1) H = − 2 + ω xi + 2 , λ = , 2 ∂x 2 (xi − xj ) 2 i=1 i 1≤i

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 33 / 60 “Statistical interaction” is repulsive at g > 1/2 and attractive at g < 1/2. High-temperature expansion of the speciﬁc heat of D-dim. ideal s excluson gas with dispersion εp = ap : CV D g − 1/2 ρD D = 1 + D/s D/s 1 − + ... , (43) N s 2 AD,s T s where ρD is the D-dim density, AD,s is some constant.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 34 / 60 High-temperature expansion of the speciﬁc heat of D-dim. ideal s excluson gas with dispersion εp = ap : CV D g − 1/2 ρD D = 1 + D/s D/s 1 − + ... , (43) N s 2 AD,s T s where ρD is the D-dim density, AD,s is some constant.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 34 / 60 Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

Equation of state of a 2D ideal gas of N particles with dispersion 2 2 µ/T εk = ~ k /2m in the limit of e 1, when the function w(x) = x + g − 1 (Wu 1994): p 2g − 1 = ρ 1 + ρ λ2 , (42) T 2 4 2 where p is pressure, T is temperature, ρ2 = N/V2 is the 2D density. This is nothing but the virial expansion. “Statistical interaction” is repulsive at g > 1/2 and attractive at g < 1/2. High-temperature expansion of the speciﬁc heat of D-dim. ideal s excluson gas with dispersion εp = ap : CV D g − 1/2 ρD D = 1 + D/s D/s 1 − + ... , (43) N s 2 AD,s T s where ρD is the D-dim density, AD,s is some constant. A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 34 / 60 Aoyama (2001) used the tight-binding model and showed that the heat capacity of the 2D lattice gas exhibits some peculiarities unlike an ordinary system having no inﬂuence on the statistics parameter.

Qin & Chen (2012) analyzed the speed of sound and compressibility of excluson gas in the harmonic trap.

Anghel et al. (2012) used the fractional exclusion statistics to calculate thermodynamic properties of the relativistic nuclear matter.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

Thermodynamics of the ideal Haldane–Wu gas was studied by several groups (Isakov et al. 1996; Joyce et al. 1996).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 35 / 60 Qin & Chen (2012) analyzed the speed of sound and compressibility of excluson gas in the harmonic trap.

Anghel et al. (2012) used the fractional exclusion statistics to calculate thermodynamic properties of the relativistic nuclear matter.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

Thermodynamics of the ideal Haldane–Wu gas was studied by several groups (Isakov et al. 1996; Joyce et al. 1996).

Aoyama (2001) used the tight-binding model and showed that the heat capacity of the 2D lattice gas exhibits some peculiarities unlike an ordinary system having no inﬂuence on the statistics parameter.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 35 / 60 Anghel et al. (2012) used the fractional exclusion statistics to calculate thermodynamic properties of the relativistic nuclear matter.

Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

Thermodynamics of the ideal Haldane–Wu gas was studied by several groups (Isakov et al. 1996; Joyce et al. 1996).

Qin & Chen (2012) analyzed the speed of sound and compressibility of excluson gas in the harmonic trap.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 35 / 60 Statistical-mechanical generalizations Haldane–Wu statistics Haldane–Wu statistics

Thermodynamics of the ideal Haldane–Wu gas was studied by several groups (Isakov et al. 1996; Joyce et al. 1996).

Qin & Chen (2012) analyzed the speed of sound and compressibility of excluson gas in the harmonic trap.

Anghel et al. (2012) used the fractional exclusion statistics to calculate thermodynamic properties of the relativistic nuclear matter.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 35 / 60 Total number of microstates:

Y Gi (Gi − γ)(Gi − 2γ) ... (Gi − (Ni − 1)γ) W = . (44) Ni ! i or

N (Gi /γ)! Wi = γ . (45) N!(Gi /γ − Ni )! One can show in a standard way that mean occupation numbers ni = Ni /Gi in this statistics are given by a simple expression: 1 ni = . (46) e(εi −µ)/T + γ

Statistical-mechanical generalizations Polychronakos statistics Polychronakos statistics

Polychronakos (1996): let the ﬁrst particle in a system can occupy one of G states, the second particle has (G − γ) states to choose, the third one has (G − 2γ) states to choose, etc.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 36 / 60 or

Statistical-mechanical generalizations Polychronakos statistics Polychronakos statistics

Polychronakos (1996): let the ﬁrst particle in a system can occupy one of G states, the second particle has (G − γ) states to choose, the third one has (G − 2γ) states to choose, etc. Total number of microstates:

Y Gi (Gi − γ)(Gi − 2γ) ... (Gi − (Ni − 1)γ) W = . (44) Ni ! i

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 36 / 60 One can show in a standard way that mean occupation numbers ni = Ni /Gi in this statistics are given by a simple expression: 1 ni = . (46) e(εi −µ)/T + γ

Statistical-mechanical generalizations Polychronakos statistics Polychronakos statistics

Y Gi (Gi − γ)(Gi − 2γ) ... (Gi − (Ni − 1)γ) W = . (44) Ni ! i or

N (Gi /γ)! Wi = γ . (45) N!(Gi /γ − Ni )!

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 36 / 60 Statistical-mechanical generalizations Polychronakos statistics Polychronakos statistics

Y Gi (Gi − γ)(Gi − 2γ) ... (Gi − (Ni − 1)γ) W = . (44) Ni ! i or

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 36 / 60 s Virial coeﬃcients of the D-dim. gas with dispersion εp = ap in the Polychronakos statistics:

P j−1 B bj (γ) = |γ| bj , if γ < 0, P j−1 F bj (γ) = γ bj , if γ > 0, (47)

B,F where bj is the jthe virial coeﬃcient of the Bose- or Fermi-system, respectively.

Statistical-mechanical generalizations Polychronakos statistics Polychronakos statistics

Acharya & Narayana Swamy (1994) considered the abovementioned expression as a simple variant of statistical-mechanical description of anyons with a correct limiting behavior in both fermionic and bosonic limits.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 37 / 60 Statistical-mechanical generalizations Polychronakos statistics Polychronakos statistics

s Virial coeﬃcients of the D-dim. gas with dispersion εp = ap in the Polychronakos statistics:

P j−1 B bj (γ) = |γ| bj , if γ < 0, P j−1 F bj (γ) = γ bj , if γ > 0, (47)

B,F where bj is the jthe virial coeﬃcient of the Bose- or Fermi-system, respectively.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 37 / 60 Mirza & Mohammadzadeh (2010) studied so called thermodynamic geometry of several fractional statistics types and reported a phenomenon similar to the Bose-condensation in a gas obeying the Polychronakos statistics. Zare et al. (2012) considered the Bose-gas on a stretched horizon of Schwarzschild and Kerr black holes using the Polychronakos statistics to model interactions in the graviton gas. With the parameter γ on the unit circle γ = eiπν, ν = 0 ÷ 1, it is possible to show that in the bosonic limit such a statistics corresponds to the Bose-gas with a spectrum containing a small dissipative part p = εp + iκp, which is linked to the statistics parameter ν: κp ' πνT εp (Rovenchak 2012).

Statistical-mechanical generalizations Polychronakos statistics Polychronakos statistics

Distribution function (46) of the Polychronakos statistics can occur in the context of q-deformed algebras.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 38 / 60 Zare et al. (2012) considered the Bose-gas on a stretched horizon of Schwarzschild and Kerr black holes using the Polychronakos statistics to model interactions in the graviton gas. With the parameter γ on the unit circle γ = eiπν, ν = 0 ÷ 1, it is possible to show that in the bosonic limit such a statistics corresponds to the Bose-gas with a spectrum containing a small dissipative part p = εp + iκp, which is linked to the statistics parameter ν: κp ' πνT εp (Rovenchak 2012).

Statistical-mechanical generalizations Polychronakos statistics Polychronakos statistics

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 38 / 60 With the parameter γ on the unit circle γ = eiπν, ν = 0 ÷ 1, it is possible to show that in the bosonic limit such a statistics corresponds to the Bose-gas with a spectrum containing a small dissipative part p = εp + iκp, which is linked to the statistics parameter ν: κp ' πνT εp (Rovenchak 2012).

Statistical-mechanical generalizations Polychronakos statistics Polychronakos statistics

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 38 / 60 Statistical-mechanical generalizations Polychronakos statistics Polychronakos statistics

Distribution function (46) of the Polychronakos statistics can occur in the context of q-deformed algebras. Mirza & Mohammadzadeh (2010) studied so called thermodynamic geometry of several fractional statistics types and reported a phenomenon similar to the Bose-condensation in a gas obeying the Polychronakos statistics. Zare et al. (2012) considered the Bose-gas on a stretched horizon of Schwarzschild and Kerr black holes using the Polychronakos statistics to model interactions in the graviton gas. With the parameter γ on the unit circle γ = eiπν, ν = 0 ÷ 1, it is possible to show that in the bosonic limit such a statistics corresponds to the Bose-gas with a spectrum containing a small dissipative part p = εp + iκp, which is linked to the statistics parameter ν: κp ' πνT εp (Rovenchak 2012).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 38 / 60 Nonextensive statistics Nonextensive statistics

Traditionally, entropy is deﬁned as the logarithm of the number of microstates:

S = ln W , (48)

It is additive:

S(A + B) = S(A) + S(B), (49) where A and B denote subsystems.

Boltzmann–Gibbs entropy can be expressed via probabilities pj of the jth state: X S = − pj ln pj . (50) j

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 39 / 60 Nonextensive Tsallis statistics (1988), generalized entropy:

W ! W 1 X X S = 1 − pq , p = 1. q ∈ (51) q q − 1 n n R n=1 n=1

Nonextensive statistics Nonextensive statistics

Additivity of the entropy can be violated for various systems, including: fractal structures; systems with long-range interactions; essentially non-Markovian processes (systems with “memory”); such approaches are applicable in social sciences and humanities (models of ﬁnancial markets, linguistic laws, etc.).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 40 / 60 Nonextensive statistics Nonextensive statistics

Nonextensive Tsallis statistics (1988), generalized entropy:

W ! W 1 X X S = 1 − pq , p = 1. q ∈ (51) q q − 1 n n R n=1 n=1

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 40 / 60 New condition of additivity (extensivity):

Sq(A + B) = Sq(A) + Sq(B) + (1 − q)Sq(A)Sq(B), (52)

that is, Sq is a nonextensive/nonadditive quantity. The q index is in fact a nonextensivity measure.

Nonextensive statistics Nonextensive statistics

In the q → 1 limit, the Boltzmann–Gibbs entropy is obtained:

q−1 (q−1) ln pn pn = e ' 1 + (q − 1) ln pn and entropy becomes

W ! W 1 X X S = 1 − pq = ... = − p ln p , q q − 1 n n n n=1 n=1 coinciding with the expected result.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 41 / 60 Nonextensive statistics Nonextensive statistics

In the q → 1 limit, the Boltzmann–Gibbs entropy is obtained:

q−1 (q−1) ln pn pn = e ' 1 + (q − 1) ln pn and entropy becomes

W ! W 1 X X S = 1 − pq = ... = − p ln p , q q − 1 n n n n=1 n=1 coinciding with the expected result. New condition of additivity (extensivity):

Sq(A + B) = Sq(A) + Sq(B) + (1 − q)Sq(A)Sq(B), (52) that is, Sq is a nonextensive/nonadditive quantity. The q index is in fact a nonextensivity measure. A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 41 / 60 In the limit of q → 1 this leads to the known Boltzmann relation S = ln W . Using the q-logarithm x 1−q − 1 ln x ≡ , ln x = ln x, (54) q 1 − q 1 one can write the Tsallis entropy in a Boltzmann-like form

Sq = lnq W . (55)

Nonextensive statistics Nonextensive statistics

As the ordinary entropy, Sq reaches maximum at equal probabilities pn = 1/W ∀n (so called Laplace principle): W 1−q − 1 S = . (53) q 1 − q

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 42 / 60 Nonextensive statistics Nonextensive statistics

As the ordinary entropy, Sq reaches maximum at equal probabilities pn = 1/W ∀n (so called Laplace principle): W 1−q − 1 S = . (53) q 1 − q In the limit of q → 1 this leads to the known Boltzmann relation S = ln W . Using the q-logarithm x 1−q − 1 ln x ≡ , ln x = ln x, (54) q 1 − q 1 one can write the Tsallis entropy in a Boltzmann-like form

Sq = lnq W . (55)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 42 / 60 Other q-exponentials appearing in some other problems involving in particular q-deformed commutators, can be deﬁned as follows:

∞ j ∞ j x X x x X j(j−1)/2 x expq x ≡ eq = , Expq x ≡ Eq = q , [j]q! [j]q! j=0 j=0

where [j]q! is the q-factorial. It is clear that x x x e1 = E1 = e . (57)

Nonextensive statistics Nonextensive statistics

The q-exponential (Tsallis q-exponential) is inverse to the q-logarithm:

1/(1−q) expq(x) = [1 + (1 − q)x] , (56) becoming an ordinary exponential as q → 1.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 43 / 60 Nonextensive statistics Nonextensive statistics

The q-exponential (Tsallis q-exponential) is inverse to the q-logarithm:

1/(1−q) expq(x) = [1 + (1 − q)x] , (56) becoming an ordinary exponential as q → 1.

Other q-exponentials appearing in some other problems involving in particular q-deformed commutators, can be deﬁned as follows:

∞ j ∞ j x X x x X j(j−1)/2 x expq x ≡ eq = , Expq x ≡ Eq = q , [j]q! [j]q! j=0 j=0 where [j]q! is the q-factorial. It is clear that x x x e1 = E1 = e . (57)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 43 / 60 with, however, q 1 1 T == q−1 , not just T = . (59) Zq β β Additional coeﬃcient in the relation between T and β complicates the description of systems using the Tsallis entropy. Some modiﬁcations of the described approach exist, known as statistics of (Tsallis–)Mendez–Plastino (1998), Curado(–Tsallis) (1991), Bashkirov (2006), etc.

Nonextensive statistics Nonextensive statistics

Applying the standard method of Lagrange multipliers, one can obtain for probabilities pn the Gibbs-like expression:

1 εn pn = expq − , (58) Zq T

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 44 / 60 Additional coeﬃcient in the relation between T and β complicates the description of systems using the Tsallis entropy. Some modiﬁcations of the described approach exist, known as statistics of (Tsallis–)Mendez–Plastino (1998), Curado(–Tsallis) (1991), Bashkirov (2006), etc.

Nonextensive statistics Nonextensive statistics

Applying the standard method of Lagrange multipliers, one can obtain for probabilities pn the Gibbs-like expression:

1 εn pn = expq − , (58) Zq T with, however, q 1 1 T == q−1 , not just T = . (59) Zq β β

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 44 / 60 Some modiﬁcations of the described approach exist, known as statistics of (Tsallis–)Mendez–Plastino (1998), Curado(–Tsallis) (1991), Bashkirov (2006), etc.

Nonextensive statistics Nonextensive statistics

Applying the standard method of Lagrange multipliers, one can obtain for probabilities pn the Gibbs-like expression:

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 44 / 60 Nonextensive statistics Nonextensive statistics

Applying the standard method of Lagrange multipliers, one can obtain for probabilities pn the Gibbs-like expression:

1 εn pn = expq − , (58) Zq T with, however, q 1 1 T == q−1 , not just T = . (59) Zq β β Additional coeﬃcient in the relation between T and β complicates the description of systems using the Tsallis entropy. Some modiﬁcations of the described approach exist, known as statistics of (Tsallis–)Mendez–Plastino (1998), Curado(–Tsallis) (1991), Bashkirov (2006), etc.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 44 / 60 Nonextensive statistics Nonextensive statistics

q-generalizations of the Fermi-Dirac and Bose–Einstein distributions are also known:

1 ni = 1 (60) {1 + (q − 1)β(εi − µ)} q−1 ± 1

1 = . expq[β(εi − µ)] ± 1

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 45 / 60 Occupation numbers of the Bose- and Fermi-like excitations in the so-called extensive incomplete statistics are: 1 ni = . (62) eq(εi −µ)/T ± 1 Nonextensive analogs are obtained is the fashion similar to the Tsallis statistics: 1 1 ni = q = {1 + (q − 1)β(εi − µ)} q−1 ± 1 expq[qβ(εi − µ)] ± 1

Nonextensive statistics Nonextensive statistics

In the so-called incomplete information theory, the following incomplete normalization ad respective entropy are considered:

W W X q X q pi = 1, Sq = − pi ln pi . (61) i=1 i=1

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 46 / 60 Nonextensive analogs are obtained is the fashion similar to the Tsallis statistics: 1 1 ni = q = {1 + (q − 1)β(εi − µ)} q−1 ± 1 expq[qβ(εi − µ)] ± 1

Nonextensive statistics Nonextensive statistics

In the so-called incomplete information theory, the following incomplete normalization ad respective entropy are considered:

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 46 / 60 Nonextensive statistics Nonextensive statistics

In the so-called incomplete information theory, the following incomplete normalization ad respective entropy are considered:

W W X q X q pi = 1, Sq = − pi ln pi . (61) i=1 i=1 Occupation numbers of the Bose- and Fermi-like excitations in the so-called extensive incomplete statistics are: 1 ni = . (62) eq(εi −µ)/T ± 1 Nonextensive analogs are obtained is the fashion similar to the Tsallis statistics: 1 1 ni = q = {1 + (q − 1)β(εi − µ)} q−1 ± 1 expq[qβ(εi − µ)] ± 1

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 46 / 60 The second virial coeﬃcients of the Bose- and Fermi-systems: 1 1 bF = + , bB = − . (65) 2 4 2 4

Connection between diﬀerent types of statistics Virial expansion Connection between diﬀerent types of statistics: Virial expansion

Virial expansion for the equation of state of a 2D system reads: p = ρ 1 + b ρ λ2 + b (ρ λ2)2 + ... , (63) T 2 2 2 3 2 where p is pressure, T is absolute temperature, ρ2 = N/V2 is a 2D density (concentration), and 2π 2 1/2 λ = ~ (64) mT is the thermal de Broglie wavelength. Factors bj are dimensionless jth virial coeﬃcients.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 47 / 60 Connection between diﬀerent types of statistics Virial expansion Connection between diﬀerent types of statistics: Virial expansion

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 47 / 60 For higher virial coeﬃcients exact analytical results are not known due the complexity of the problem of N ≥ 3 anyons

Exact symmetry relation for the third virial coeﬃcient:

b3(α) = b3(1 − α). (67)

Connection between diﬀerent types of statistics Virial expansion Virial expansion

The second virial coeﬃcient of the ideal anyon gas: 1 b (α) = − (1 − 4α + 2α2). (66) 2 4

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 48 / 60 Exact symmetry relation for the third virial coeﬃcient:

b3(α) = b3(1 − α). (67)

Connection between diﬀerent types of statistics Virial expansion Virial expansion

The second virial coeﬃcient of the ideal anyon gas: 1 b (α) = − (1 − 4α + 2α2). (66) 2 4

For higher virial coeﬃcients exact analytical results are not known due the complexity of the problem of N ≥ 3 anyons

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 48 / 60 Connection between diﬀerent types of statistics Virial expansion Virial expansion

The second virial coeﬃcient of the ideal anyon gas: 1 b (α) = − (1 − 4α + 2α2). (66) 2 4

For higher virial coeﬃcients exact analytical results are not known due the complexity of the problem of N ≥ 3 anyons

Exact symmetry relation for the third virial coeﬃcient:

b3(α) = b3(1 − α). (67)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 48 / 60 Ideal 2D gas obeying the Gentile statistics with maximal occupation s ≥ 2 has the 2nd virial coeﬃcient 1 bG = bB = − , (68) 2 2 4 which is impossible to link with that of anyons 1 2 b2(α) = − 4 (1 − 4α + 2α ), except for the trivial case α = 0.

Connection between diﬀerent types of statistics Virial expansion Virial expansion

Expressions for virial coeﬃcients can be applied to establish correspondences between anyonic statistics and other types of fractional statistics.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 49 / 60 Connection between diﬀerent types of statistics Virial expansion Virial expansion

Expressions for virial coeﬃcients can be applied to establish correspondences between anyonic statistics and other types of fractional statistics.

Ideal 2D gas obeying the Gentile statistics with maximal occupation s ≥ 2 has the 2nd virial coeﬃcient 1 bG = bB = − , (68) 2 2 4 which is impossible to link with that of anyons 1 2 b2(α) = − 4 (1 − 4α + 2α ), except for the trivial case α = 0.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 49 / 60 Comparing it to that of anyons, the following relation between the parameters of statistics g and α is obtained:

g = 2α − α2. (70)

Connection between diﬀerent types of statistics Virial expansion Virial expansion

The 2nd virial coeﬃcient of the ideal 2D Haldane–Wu gas equals 1 bHW = (2g − 1). (69) 2 4

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 50 / 60 Connection between diﬀerent types of statistics Virial expansion Virial expansion

The 2nd virial coeﬃcient of the ideal 2D Haldane–Wu gas equals 1 bHW = (2g − 1). (69) 2 4 Comparing it to that of anyons, the following relation between the parameters of statistics g and α is obtained:

g = 2α − α2. (70)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 50 / 60 However, none of the abovementioned results allows establishing a complete correspondence of anyons neither with the Haldane–Wu statistics nor with the Polychronakos statistics: the 3rd virial coeﬃcients are diﬀerent.

Connection between diﬀerent types of statistics Virial expansion Virial expansion

The 2nd virial coeﬃcient of the ideal 2D Polychronakos gas 1 bP = − |γ|, (71) 2 4 (boson-like statistics type for γ < 0) leads to the following connection between γ and α:

γ = 4α − 2α2 − 1. (72)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 51 / 60 Connection between diﬀerent types of statistics Virial expansion Virial expansion

The 2nd virial coeﬃcient of the ideal 2D Polychronakos gas 1 bP = − |γ|, (71) 2 4 (boson-like statistics type for γ < 0) leads to the following connection between γ and α:

γ = 4α − 2α2 − 1. (72)

However, none of the abovementioned results allows establishing a complete correspondence of anyons neither with the Haldane–Wu statistics nor with the Polychronakos statistics: the 3rd virial coeﬃcients are diﬀerent.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 51 / 60 In the high-T limit, the equation of state in the Gentile statistics contains the Bose-like correction only, which is not the case for the Haldane–Wu or Polychronakos statistics.

Approximate correspondence can be achieved on the basis of the Fermi level analog, but this cannot be extended to the whole temperature domain.

In a certain approximation, correspondence between the Haldane–Wu and Gentile statistics can be seen from the behavior of the occupation numbers.

Connection between diﬀerent types of statistics Virial expansion Connection between diﬀerent types of statistics

No complete analogy can be established between the Gentile, Haldane–Wu, and Polychronakos statistics. In particular:

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 52 / 60 Approximate correspondence can be achieved on the basis of the Fermi level analog, but this cannot be extended to the whole temperature domain.

In a certain approximation, correspondence between the Haldane–Wu and Gentile statistics can be seen from the behavior of the occupation numbers.

Connection between diﬀerent types of statistics Virial expansion Connection between diﬀerent types of statistics

No complete analogy can be established between the Gentile, Haldane–Wu, and Polychronakos statistics. In particular:

In the high-T limit, the equation of state in the Gentile statistics contains the Bose-like correction only, which is not the case for the Haldane–Wu or Polychronakos statistics.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 52 / 60 In a certain approximation, correspondence between the Haldane–Wu and Gentile statistics can be seen from the behavior of the occupation numbers.

Connection between diﬀerent types of statistics Virial expansion Connection between diﬀerent types of statistics

No complete analogy can be established between the Gentile, Haldane–Wu, and Polychronakos statistics. In particular:

In the high-T limit, the equation of state in the Gentile statistics contains the Bose-like correction only, which is not the case for the Haldane–Wu or Polychronakos statistics.

Approximate correspondence can be achieved on the basis of the Fermi level analog, but this cannot be extended to the whole temperature domain.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 52 / 60 Connection between diﬀerent types of statistics Virial expansion Connection between diﬀerent types of statistics

No complete analogy can be established between the Gentile, Haldane–Wu, and Polychronakos statistics. In particular:

In the high-T limit, the equation of state in the Gentile statistics contains the Bose-like correction only, which is not the case for the Haldane–Wu or Polychronakos statistics.

Approximate correspondence can be achieved on the basis of the Fermi level analog, but this cannot be extended to the whole temperature domain.

In a certain approximation, correspondence between the Haldane–Wu and Gentile statistics can be seen from the behavior of the occupation numbers.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 52 / 60 For deﬁniteness, we will restrict ourselves to the Bose statistics.

Let the analyzed system has spectrum εj and Gj is the degeneration of the jth level. Chemical potential of the respective Bose-system µB is linked to the number of particles N and temperature T as follows:

X Gj N = . (73) e(εj −µB)/T − 1 j

Connection between diﬀerent types of statistics Expansion over statistics parameters Expansion over statistics parameters

Let us consider small deviations from some traditional statistics (Bose or Fermi).

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 53 / 60 Let the analyzed system has spectrum εj and Gj is the degeneration of the jth level. Chemical potential of the respective Bose-system µB is linked to the number of particles N and temperature T as follows:

X Gj N = . (73) e(εj −µB)/T − 1 j

Connection between diﬀerent types of statistics Expansion over statistics parameters Expansion over statistics parameters

Let us consider small deviations from some traditional statistics (Bose or Fermi).

For deﬁniteness, we will restrict ourselves to the Bose statistics.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 53 / 60 Connection between diﬀerent types of statistics Expansion over statistics parameters Expansion over statistics parameters

Let us consider small deviations from some traditional statistics (Bose or Fermi).

For deﬁniteness, we will restrict ourselves to the Bose statistics.

X Gj N = . (73) e(εj −µB)/T − 1 j

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 53 / 60 Chemical potential of such a system µP is deﬁned by

X Gj N = , (74) e(εj −µP)/T − 1 + a j

and it can be written using a small deviation from that of the Bose-system:

µP = µB + ∆µP. (75)

Connection between diﬀerent types of statistics Expansion over statistics parameters Expansion over statistics parameters

Let us now consider a system obeying the Polychronakos statistics with the parameter γ = a − 1, where a → 0.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 54 / 60 Connection between diﬀerent types of statistics Expansion over statistics parameters Expansion over statistics parameters

Let us now consider a system obeying the Polychronakos statistics with the parameter γ = a − 1, where a → 0.

Chemical potential of such a system µP is deﬁned by

X Gj N = , (74) e(εj −µP)/T − 1 + a j and it can be written using a small deviation from that of the Bose-system:

µP = µB + ∆µP. (75)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 54 / 60 Taking into account (73), one obtains:

∆µP P X Gj = a , where P = 2 , (77) T N + P (εj −µB)/T j e − 1

Connection between diﬀerent types of statistics Expansion over statistics parameters Expansion over statistics parameters

Expand further the expression under summation in Eq. (74) into series wrt small a and ∆µP to the linear corrections:

X Gj X Gj N = = + e(εj −µP)/T − 1 + a e(εj −µB)/T − 1 j j G ∆µ X j P (εj −µB)/T + 2 e − a . (76) (εj −µB)/T T j e − 1

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 55 / 60 Connection between diﬀerent types of statistics Expansion over statistics parameters Expansion over statistics parameters

Expand further the expression under summation in Eq. (74) into series wrt small a and ∆µP to the linear corrections:

X Gj X Gj N = = + e(εj −µP)/T − 1 + a e(εj −µB)/T − 1 j j G ∆µ X j P (εj −µB)/T + 2 e − a . (76) (εj −µB)/T T j e − 1

Taking into account (73), one obtains:

∆µP P X Gj = a , where P = 2 , (77) T N + P (εj −µB)/T j e − 1

In a similar fashion, one can deﬁne the correction to the chemical potential in a Bose-system analog under within the Tsallis approach:

X Gj X Gj N = = + (78) (εj −µTs)/T e(εj −µB)/T − 1 j eq − 1 j ( ) (εj −µB)/T 2 X Gj e ∆µTs εj − µB q − 1 + 2 − , (εj −µB)/T T T 2 j e − 1 where q → 1 and chemical potential

µTs = µB + ∆µTs. (79)

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 56 / 60 Connection between diﬀerent types of statistics Expansion over statistics parameters Expansion over statistics parameters

Similar expansions can be written for a Bose-system with the spectrum εj + ∆εj , where a small correction ∆εj is, for instance, caused by interactions. For chemical potential µ we have µB:

µ = µB + ∆µ, (80) while

X Gj X Gj N = = + e(εj +∆εj −µ)/T − 1 e(εj −µB)/T − 1 j j

(εj −µB)/T X Gj e ∆µ − ∆εj + 2 . (81) (εj −µB)/T T j e − 1

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 57 / 60 To ﬁnd such correspondences one can require, e.g., that energies in diﬀerent systems in the respective statistics coincide, that is: X E = j Gj nj = EB + ∆E, (82) j

where EB is the energy of the Bose-system with spectrum εj : X εj Gj EB = , (83) e(εj −µB)/T − 1 j

here, j is the excitation spectrum (equal to εj or εj + ∆εj in the considered examples), and ∆E is a correction.

Connection between diﬀerent types of statistics Expansion over statistics parameters Expansion over statistics parameters

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 58 / 60 Connection between diﬀerent types of statistics Expansion over statistics parameters Expansion over statistics parameters

It means that, to a certain degree, an interacting Bose-system can be modeled by systems obeying various types of fractional statistics with parameters being defined by the spectrum correction ∆εj . To find such correspondences one can require, e.g., that energies in different systems in the respective statistics coincide, that is: X E = j Gj nj = EB + ∆E, (82) j where EB is the energy of the Bose-system with spectrum εj : X εj Gj EB = , (83) e(εj −µB)/T − 1 j here, j is the excitation spectrum (equal to εj or εj + ∆εj in the considered examples), and ∆E is a correction.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 58 / 60 Main approaches to define a fractional (intermediate) statistics generalizing conventional quantum Bose–Einstein and Fermi–Dirac distributions are analyzed. Introductory information is given on anyons and some q-deformed algebras of creation–annihilation operators are briefly reviewed. Special attention is paid to statistical-mechanical approaches by Gentile, Haldane–Wu, and Polychronakos. Nonextensive/nonadditive generalizations of the Boltzmann–Gibbs entropy and respective fractional statistics are considered as well. Some possibilities to find connections between different types of statistics are briefly analyzed.

Summary Summary

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 59 / 60 Introductory information is given on anyons and some q-deformed algebras of creation–annihilation operators are briefly reviewed. Special attention is paid to statistical-mechanical approaches by Gentile, Haldane–Wu, and Polychronakos. Nonextensive/nonadditive generalizations of the Boltzmann–Gibbs entropy and respective fractional statistics are considered as well. Some possibilities to find connections between different types of statistics are briefly analyzed.

Summary Summary

Main approaches to deﬁne a fractional (intermediate) statistics generalizing conventional quantum Bose–Einstein and Fermi–Dirac distributions are analyzed.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 59 / 60 Special attention is paid to statistical-mechanical approaches by Gentile, Haldane–Wu, and Polychronakos. Nonextensive/nonadditive generalizations of the Boltzmann–Gibbs entropy and respective fractional statistics are considered as well. Some possibilities to find connections between different types of statistics are briefly analyzed.

Summary Summary

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 59 / 60 Nonextensive/nonadditive generalizations of the Boltzmann–Gibbs entropy and respective fractional statistics are considered as well. Some possibilities to find connections between different types of statistics are briefly analyzed.

Summary Summary

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 59 / 60 Some possibilities to find connections between different types of statistics are briefly analyzed.

Summary Summary

Main approaches to deﬁne a fractional (intermediate) statistics generalizing conventional quantum Bose–Einstein and Fermi–Dirac distributions are analyzed. Introductory information is given on anyons and some q-deformed algebras of creation–annihilation operators are brieﬂy reviewed. Special attention is paid to statistical-mechanical approaches by Gentile, Haldane–Wu, and Polychronakos. Nonextensive/nonadditive generalizations of the Boltzmann–Gibbs entropy and respective fractional statistics are considered as well.

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 59 / 60 Summary Summary

A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 59 / 60 Summary

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A. Rovenchak (Lviv Univ.) Fractional statistics: A retrospective Anyon Physics . . . 60 / 60