DOCSLIB.ORG

Contents 1 the Free Enthalpy

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Contents 1 the Free Enthalpy PHY304 - Statistical Mechanics Spring Semester 2021 Dr. Anosh Joseph, IISER Mohali LECTURE 09 Thursday, January 21, 2021 (Note: This is an online lecture due to COVID-19 interruption.) Contents 1 The Free Enthalpy 1 2 The Grand Potential 4 2.1 Transformation of All Variables . .6 3 Maxwell Relations 6 1 The Free Enthalpy Consider a system with given temperature T and pressure p. We can then perform the Legendre transformation of the internal energy U(S; V; N; ··· ) with respect to two variables S and V . We have G = U − TS + pV: (1) The function G is also a thermodynamic potential. This is called the free enthalpy. It was introduced by Gibbs in 1875. It is also known as the Gibb’s potential. Taking the total differential of G we have dG = dU − T dS − SdT + pdV + V dp = −SdT + V dp + µdN + ··· : (2) From this we see that G depends only on T , p, and N. PHY304 - Statistical Mechanics Spring Semester 2021 We can obtain the equations of state from G. We have @G −S = ; (3) @T p;N;··· @G V = ; (4) @p T;N;··· @G µ = ; (5) @N T;p;··· . : Upon using the Euler’s equation U = TS − pV + µN; (6) we have G = U − TS + pV = µN (7) We see that G is proportional to the particle number. Thus we can say that the free enthalpy per particle is identical to the chemical potential. Note that this statement is valid only for systems containing one kind (species) of particles, which cannot exchange other forms of energy with their surroundings. To understand the meaning of the free enthalpy let us consider an isolated system consisting of the isothermal and isobaric system immersed in a heat bath. See Fig. 1. Then we have dStot = dSsys + dSbath ≥ 0: (8) The equality sign holds for reversible processes. The greater-than-or-equal-to sign for irreversible processes. For reversible processes we have 1 dS = −dS = − (dU + pdV − δW rev ) : (9) bath sys T sys sys other For irreversible processes we must have rev T dStot = T dSsys − dUsys − pdVsys + δWother ≥ 0: (10) We can write the above two equations as rev irr dGsys = d(U − TS + pV ) = δWother ≤ δWother: (11) sys 2 / 9 PHY304 - Statistical Mechanics Spring Semester 2021 F ⃗ = const . T, p, N heat bath T Figure 1: Isothermal and isobaric system. This tells us that the change in free enthalpy is just the work performed by the system in an isothermal, isobaric reversible process, without volume work against the constant external pressure. We see that in an isothermal, isobaric system which is left to its own, irreversible processes happen until a minimum of the free enthalpy is reached, dG = 0; (12) G = Gmin: (13) Similar to enthalpy, the free enthalpy is of great importance for chemistry. This is useful in the case of many types of fuel cells or batteries. There, chemical reactions happen slowly under constant (atmospheric) pressure maintaining thermal equilibrium. We can directly calculate the electrical work obtainable from a battery as the difference of the free enthalpies in the final and initial states. As an example, the free enthalpy of the ideal gas is G(T; p; N) = Nµ(T; p): (14) The free enthalpy and its derivative with respect to temperature can be related to the enthalpy 3 / 9 PHY304 - Statistical Mechanics Spring Semester 2021 of the system @G H = G − T @T p @(G=T ) = −T 2 : (15) @T p The above relation is known as the Gibbs-Helmholtz equation. If the isothermal and isobaric system consists of several chemical components (particle species) we then have G = µiNi: (16) The equilibrium condition takes the form dG = µidNi = 0: (17) Note that we can apply the above considerations to systems containing elementary particles in a hot star (or plasma), instead of a chemical reaction. 2 The Grand Potential Let us consider systems where the chemical potential is given as a state variable, instead of the particle number N. The chemical potential can be fixed by a particle bath. The exchange of particles with a particle reservoir leads to a constant chemical potential. Let us transform the internal energy in the state variables S and N to the new variables T and µ Φ = U − TS − µN: (18) This potential is called the grand potential. Note that this potential is of great importance for the statistical treatment of thermodynamic problems. The total differential has the form dΦ = dU − T dS − SdT − µdN − Ndµ, = −SdT − pdV − Ndµ. (19) 4 / 9 PHY304 - Statistical Mechanics Spring Semester 2021 The equations of state are @Φ −S = ; (20) @T V,µ @Φ −p = ; (21) @V T;µ @Φ −N = : (22) @µ T;V Upon using the Euler’s equation U = TS − pV + µN we see that Φ = −pV: (23) This potential becomes useful when we are dealing with isothermal systems with a fixed chemical potential. See Fig. 2. particle reservoir μ T, V, μ heat bath T Figure 2: Isothermal system with fixed chemical potential. If we combine the heat bath and the system under consideration to a total isolated system, then we must have dStot = dSsys + dSbath ≥ 0: (24) In the reversible case we have rev T dSbath = −T dSsys = − (dUsys − µdNsys − δWother) : (25) 5 / 9 PHY304 - Statistical Mechanics Spring Semester 2021 Rearranging terms rev irr dUsys − T dSsys − µdNsys = δWother ≤ δWother: (26) rev irr Here δWother and δWother correspond respectively, to the work performed reversibly and irre- versibly, by the system, without the explicitly considered chemical energy. At constant temperature and constant chemical potential this is equivalent to rev irr dΦ = d(U − TS − µN) = δWother ≤ δWother: (27) If we leave the system on its own without performing work, δW = 0, it strives for a minimum of the grand potential dΦ ≤ 0; (28) which is achieved in equilibrium dΦ = 0; (29) Φ = Φmin: (30) 2.1 Transformation of All Variables Let us consider the effect of transforming all variables in U to the new variables T; p; µ. The Legendre transformation is Ψ = U − TS + pV − µN: (31) Upon using the Euler’s equation U = TS − pV + µN; we get Ψ ≡ 0: (32) That is, this potential vanishes identically. This tells us that the simultaneous transformation of all the variables has no relevance. 3 Maxwell Relations The thermodynamic potentials U; F; H; G and Φ are all state functions. That is, they have exact differentials. From the above fact we can derive a variety of relations among themselves. 6 / 9 PHY304 - Statistical Mechanics Spring Semester 2021 We have dU = T dS − pdV + µdN @U @U @U = dS + dV + dN: (33) @S @V @N V;N S;N S;V Since 0 1 0 1 @ @U @ @U = ; (34) @V @ @S A @S @@V A V;N S;N S;N V;N we have @T @p = − : (35) @V @S S;N V;N We can obtain many relations in this way, that may allow for the calculation of unknown quantities from known quantities. From Eq. (33) we have @T @p = − ; (36) @V @S S;N V;N @T @µ = ; (37) @N @S S;V V;N @p @µ − = : (38) @N @V S;V S;N From free energy we can obtain a similar set of equations dF = −SdT − pdV + µdN; (39) @S @p − = − ; (40) @V @T T;N V;N @S @µ − = ; (41) @N @T T;V V;N @p @µ − = : (42) @N @V T;V T;N From enthalpy we get dH = T dS + V dp + µdN; (43) 7 / 9 PHY304 - Statistical Mechanics Spring Semester 2021 @T @V = − ; (44) @p @S S;N p;N @T @µ = ; (45) @N @S S;p p;N @V @µ = : (46) @N @p S;p S;N For the free enthalpy we have dG = −SdT + V dp + µdN; (47) @S @V − = ; (48) @p @T T;N p;N @S @µ − = ; (49) @N @T T;p p;N @V @µ = : (50) @N @p T;p T;N For the grand potential we have dΦ = −SdT − pdV − Ndµ, (51) @S @p = ; (52) @V @T T,µ V,µ @S @N = ; (53) @µ @T T;V V,µ @p @N = : (54) @µ @V T;V T,µ The above set of relations are called Maxwell relations. If there are even more state variables (magnetic field, magnetic dipole moment, etc.) we need to add more relations. There exists a simple device which allows for a quick overlook of the potentials and their variables, and which yields the Maxwell relations. 8 / 9 PHY304 - Statistical Mechanics Spring Semester 2021 This device is the thermodynamic rectangle. See Fig. 3. F V T U G S p H Figure 3: Thermodynamic rectangle at constant N. The thermodynamic rectangle was conceived especially for systems with constant particle num- ber and without further state variables. The variables V; T; p and S, which are the only possible quantities at constant particle number, form the corners of this rectangle. Along the edges we have the potentials, which depend on the variables at the corresponding corners. With this way of presentation we can easily read off the partial derivatives. The derivative of a potential with respect to a variable (corner) is just given by the variable at the diagonally opposite corner. The arrows in the diagonals determine the sign. As an example, we have @F=@V = −p. The minus sign occurs because the direction V ! p is opposite to the direction of the flow. We also have @G=@p = +V . We can also easily read off the Maxwell relations. Derivatives of variables along an edge of the rectangle (e.g., @V=@S), at constant variable in the diagonally opposite corner (here p), are just equal to the corresponding derivative along the other side, i.e., in this case @T=@p .
Load more

Recommended publications
  • Grand-Canonical Ensemble
    PHYS4006: Thermal and Statistical Physics Lecture Notes (Unit - IV) Open System: Grand-canonical Ensemble Dr. Neelabh Srivastava (Assistant Professor) Department of Physics Programme: M.Sc. Physics Mahatma Gandhi Central University Semester: 2nd Motihari-845401, Bihar E-mail: [email protected] • In microcanonical ensemble, each system contains same fixed energy as well as same number of particles. Hence, the system dealt within this ensemble is a closed isolated system. • With microcanonical ensemble, we can not deal with the systems that are kept in contact with a heat reservoir at a given temperature. 2 • In canonical ensemble, the condition of constant energy is relaxed and the system is allowed to exchange energy but not the particles with the system, i.e. those systems which are not isolated but are in contact with a heat reservoir. • This model could not be applied to those processes in which number of particle varies, i.e. chemical process, nuclear reactions (where particles are created and destroyed) and quantum process. 3 • So, for the method of ensemble to be applicable to such processes where number of particles as well as energy of the system changes, it is necessary to relax the condition of fixed number of particles. 4 • Such an ensemble where both the energy as well as number of particles can be exchanged with the heat reservoir is called Grand Canonical Ensemble. • In canonical ensemble, T, V and N are independent variables. Whereas, in grand canonical ensemble, the system is described by its temperature (T),volume (V) and chemical potential (μ). 5 • Since, the system is not isolated, its microstates are not equally probable.
    [Show full text]
  • The Grand Canonical Ensemble
    University of Central Arkansas The Grand Canonical Ensemble Stephen R. Addison Directory ² Table of Contents ² Begin Article Copyright °c 2001 [email protected] Last Revision Date: April 10, 2001 Version 0.1 Table of Contents 1. Systems with Variable Particle Numbers 2. Review of the Ensembles 2.1. Microcanonical Ensemble 2.2. Canonical Ensemble 2.3. Grand Canonical Ensemble 3. Average Values on the Grand Canonical Ensemble 3.1. Average Number of Particles in a System 4. The Grand Canonical Ensemble and Thermodynamics 5. Legendre Transforms 5.1. Legendre Transforms for two variables 5.2. Helmholtz Free Energy as a Legendre Transform 6. Legendre Transforms and the Grand Canonical Ensem- ble 7. Solving Problems on the Grand Canonical Ensemble Section 1: Systems with Variable Particle Numbers 3 1. Systems with Variable Particle Numbers We have developed an expression for the partition function of an ideal gas. Toc JJ II J I Back J Doc Doc I Section 2: Review of the Ensembles 4 2. Review of the Ensembles 2.1. Microcanonical Ensemble The system is isolated. This is the ¯rst bridge or route between mechanics and thermodynamics, it is called the adiabatic bridge. E; V; N are ¯xed S = k ln ­(E; V; N) Toc JJ II J I Back J Doc Doc I Section 2: Review of the Ensembles 5 2.2. Canonical Ensemble System in contact with a heat bath. This is the second bridge between mechanics and thermodynamics, it is called the isothermal bridge. This bridge is more elegant and more easily crossed. T; V; N ¯xed, E fluctuates.
    [Show full text]
  • Quantum Statistics (Quantenstatistik)
    Chapter 7 Quantum Statistics (Quantenstatistik) 7.1 Density of states of massive particles Our goal: discover what quantum effects bring into the statistical description of systems. Formulate conditions under which QM-specific effects are not essential and classical descriptions can be used. Consider a single particle in a box with infinitely high walls U ε3 ε2 Figure 7.1: Levels of a single particle in a 3D box. At the ε1 box walls the potential is infinite and the wave-function is L zero. A one-particle Hamiltonian has only the momentum p^2 ~2 H^ = = − r2 (7.1) 2m 2m The infinite potential at the box boundaries imposes the zero-wave-function boundary condition. Hence the eigenfunctions of the Hamiltonian are k = sin(kxx) sin(kyy) sin(kzz) (7.2) ~ 2π + k = ~n; ~n = (n1; n2; n3); ni 2 Z (7.3) L ( ) ~ ~ 2π 2 ~p = ~~k; ϵ = k2 = ~n2;V = L3 (7.4) 2m 2m L 4π2~2 ) ϵ = ~n2 (7.5) 2mV 2=3 Let us introduce spin s (it takes into account possible degeneracy of the energy levels) and change the summation over ~n to an integration over ϵ X X X X X X Z X Z 1 ··· = ··· = · · · ∼ d3~n ··· = dn4πn2 ::: (7.6) 0 ν s ~k s ~n s s 45 46 CHAPTER 7. QUANTUM STATISTICS (QUANTENSTATISTIK) (2mϵ)1=2 V 1=3 1 1 V m3=2 n = V 1=3 ; dn = 2mdϵ, 4πn2dn = p ϵ1=2dϵ (7.7) 2π~ 2π~ 2 (2mϵ)1=2 2π2~3 X = gs = 2s + 1 spin-degeneracy (7.8) s Z X g V m3=2 1 ··· = ps dϵϵ1=2 ::: (7.9) 2~3 ν 2π 0 Number of states in a given energy range is higher at high energies than at low energies.
    [Show full text]
  • Singularity Development and Supersymmetry in Holography
    Singularity development and supersymmetry in holography Alex Buchel Department of Applied Mathematics Department of Physics and Astronomy University of Western Ontario London, Ontario N6A 5B7, Canada Perimeter Institute for Theoretical Physics Waterloo, Ontario N2J 2W9, Canada Abstract We study the effects of supersymmetry on singularity development scenario in hologra- phy presented in [1] (BBL). We argue that the singularity persists in a supersymmetric extension of the BBL model. The challenge remains to find a string theory embedding of the singularity mechanism. May 23, 2017 arXiv:1705.08560v1 [hep-th] 23 May 2017 Contents 1 Introduction 2 2 DG model in microcanonical ensemble 4 2.1 Grandcanonicalensemble(areview) . 6 2.2 Microcanonicalensemble .......................... 8 2.3 DynamicsofDG-Bmodel ......................... 10 2.3.1 QNMs and linearized dynamics of DG-B model . 13 2.3.2 Fully nonlinear evolution of DG-B model . 15 3 Supersymmetric extension of BBL model 16 3.1 PhasediagramandQNMs ......................... 18 3.2 DynamicsofsBBLhorizons ........................ 21 3.2.1 Dynamics of symmetric sBBL sector and its linearized symmetry breakingfluctuations . .. .. 23 3.2.2 UnstablesBBLdynamics. 25 4 Conclusions 27 A Numerical setup 29 A.1 DG-Bmodel................................. 29 A.2 sBBLmodel................................. 30 1 Introduction Horizons are ubiquitous in holographic gauge theory/string theory correspondence [2,3]. Static horizons are dual to thermal states of the boundary gauge theory [4], while their long-wavelength near-equilibrium dynamics encode the effective boundary hydrodynamics of the theory [5]. Typically, dissipative effects in the hydrodynamics (due to shear and bulk viscosities) lead to an equilibration of a gauge theory state — a slightly perturbed horizon in a dual gravitational description settles to an equi- librium configuration.
    [Show full text]
  • Ideal Quantum Gases from Statistical Physics Using Mathematica © James J
    Ideal Quantum Gases from Statistical Physics using Mathematica © James J. Kelly, 1996-2002 The indistinguishability of identical particles has profound effects at low temperatures and/or high density where quantum mechanical wave packets overlap appreciably. The occupation representation is used to study the statistical mechanics and thermodynamics of ideal quantum gases satisfying Fermi- Dirac or Bose-Einstein statistics. This notebook concentrates on formal and conceptual developments, liberally quoting more technical results obtained in the auxiliary notebooks occupy.nb, which presents numerical methods for relating the chemical potential to density, and fermi.nb and bose.nb, which study the mathematical properties of special functions defined for Ferm-Dirac and Bose-Einstein systems. The auxiliary notebooks also contain a large number of exercises. Indistinguishability In classical mechanics identical particles remain distinguishable because it is possible, at least in principle, to label them according to their trajectories. Once the initial position and momentum is determined for each particle with the infinite precision available to classical mechanics, the swarm of classical phase points moves along trajectories which also can in principle be determined with absolute certainty. Hence, the identity of the particle at any classical phase point is connected by deterministic relationships to any initial condition and need not ever be confused with the label attached to another trajectory. Therefore, in classical mechanics even identical particles are distinguishable, in principle, even though we must admit that it is virtually impossible in practice to integrate the equations of motion for a many-body system with sufficient accuracy. Conversely, in quantum mechanics identical particles are absolutely indistinguishable from one another.
    [Show full text]
  • Holographic Viscoelastic Hydrodynamics
    Published for SISSA by Springer Received: May 23, 2018 Revised: March 3, 2019 Accepted: March 11, 2019 Published: March 25, 2019 Holographic viscoelastic hydrodynamics JHEP03(2019)146 Alex Buchela;b;c and Matteo Baggiolid aDepartment of Applied Mathematics, Western University, Middlesex College Room 255, 1151 Richmond Street, London, Ontario, N6A 5B7 Canada bThe Department of Physics and Astronomy, PAB Room 138, 1151 Richmond Street, London, Ontario, N6A 3K7 Canada cPerimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, ON, N2L 2Y5 Canada dCrete Center for Theoretical Physics, Physics Department, University of Crete, P.O. Box 2208, Heraklion, Crete, 71003 Greece E-mail: [email protected], [email protected] Abstract: Relativistic fluid hydrodynamics, organized as an effective field theory in the velocity gradients, has zero radius of convergence due to the presence of non-hydrodynamic excitations. Likewise, the theory of elasticity of brittle solids, organized as an effective field theory in the strain gradients, has zero radius of convergence due to the process of the thermal nucleation of cracks. Viscoelastic materials share properties of both fluids and solids. We use holographic gauge theory/gravity correspondence to study all order hydrodynamics of relativistic viscoelastic media. Keywords: Gauge-gravity correspondence, Holography and condensed matter physics (AdS/CMT), AdS-CFT Correspondence, Holography and quark-gluon plasmas ArXiv ePrint: 1805.06756 Open Access, c The Authors. https://doi.org/10.1007/JHEP03(2019)146
    [Show full text]
  • Jhep08(2017)134
    Published for SISSA by Springer Received: May 27, 2017 Revised: July 13, 2017 Accepted: August 14, 2017 Published: August 29, 2017 JHEP08(2017)134 Singularity development and supersymmetry in holography Alex Buchel Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada Department of Physics and Astronomy, University of Western Ontario, London, Ontario N6A 5B7, Canada Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9, Canada E-mail: [email protected] Abstract: We study the effects of supersymmetry on singularity development scenario in holography presented in [1] (BBL). We argue that the singularity persists in a super- symmetric extension of the BBL model. The challenge remains to find a string theory embedding of the singularity mechanism. Keywords: Black Holes in String Theory, Gauge-gravity correspondence, Spacetime Sin- gularities ArXiv ePrint: 1705.08560 Open Access, c The Authors. https://doi.org/10.1007/JHEP08(2017)134 Article funded by SCOAP3. Contents 1 Introduction 1 2 DG model in microcanonical ensemble 3 2.1 Grand canonical ensemble (a review) 4 2.2 Microcanonical ensemble 6 2.3 Dynamics of DG-B model 7 JHEP08(2017)134 2.3.1 QNMs and linearized dynamics of DG-B model 10 2.3.2 Fully nonlinear evolution of DG-B model 12 3 Supersymmetric extension of BBL model 13 3.1 Phase diagram and QNMs 14 3.2 Dynamics of sBBL horizons 17 3.2.1 Dynamics of symmetric sBBL sector and its linearized symmetry breaking fluctuations 18 3.2.2 Unstable sBBL dynamics 20 4 Conclusions 22 A Numerical setup 23 A.1 DG-B model 23 A.2 sBBL model 24 1 Introduction Horizons are ubiquitous in holographic gauge theory/string theory correspondence [2, 3].
    [Show full text]
  • 4. Classical Phase Space 4.1. Phase Space and Probability Density
    4. Classical phase space quantum states in the part of the space under consideration. 4.1. Phase space and probability density The ensemble or statistical set consists, at a given We consider a system of N particles in a d-dimensional moment, of all those phase space points which correspond space. Canonical coordinates and momenta to a given macroscopic system. Corresponding to a macro(scopic) state of the system q = (q1,...,qdN ) there are thus a set of micro(scopic) states which belong p = (p1,...,pdN ) to the ensemble with the probability ρ(P ) dΓ. ρ(P ) is the probability density which is normalized to unity: determine exactly the microscopic state of the system. The phase space is the 2dN-dimensional space (p, q) , dΓ ρ(P )=1, whose every point P = (p, q) corresponds to a possible{ } Z state of the system. A trajectory is such a curve in the phase space along and it gives the local density of points in (q,p)-space at which the point P (t) as a function of time moves. time t. Trajectories are determined by the classical equations of The statistical average, or the ensemble expectation motion value, of a measurable quantity f = f(P ) is dq ∂H i = f = dΓ f(P )ρ(P ). dt ∂p h i i Z dp ∂H i = , We associate every phase space point with the velocity dt ∂q − i field ∂H ∂H where V = (q, ˙ p˙)= , . ∂p − ∂q H = H(q ,...,q ,p ,...,p ,t) 1 dN 1 dN The probability current is then Vρ.
    [Show full text]
  • Statistical Physics University of Cambridge Part II Mathematical Tripos
    Preprint typeset in JHEP style - HYPER VERSION Statistical Physics University of Cambridge Part II Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/statphys.html [email protected] { 1 { Recommended Books and Resources • Reif, Fundamentals of Statistical and Thermal Physics A comprehensive and detailed account of the subject. It's solid. It's good. It isn't quirky. • Kardar, Statistical Physics of Particles A modern view on the subject which offers many insights. It's superbly written, if a little brief in places. A companion volume, \The Statistical Physics of Fields" covers aspects of critical phenomena. Both are available to download as lecture notes. Links are given on the course webpage • Landau and Lifshitz, Statistical Physics Russian style: terse, encyclopedic, magnificent. Much of this book comes across as remarkably modern given that it was first published in 1958. • Mandl, Statistical Physics This is an easy going book with very clear explanations but doesn't go into as much detail as we will need for this course. If you're struggling to understand the basics, this is an excellent place to look. If you're after a detailed account of more advanced aspects, you should probably turn to one of the books above. • Pippard, The Elements of Classical Thermodynamics This beautiful little book walks you through the rather subtle logic of classical ther- modynamics. It's very well done. If Arnold Sommerfeld had read this book, he would have understood thermodynamics the first time round.
    [Show full text]
  • Ensemble and Trajectory Thermodynamics: a Brief Introduction C
    Physica A 418 (2015) 6–16 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Ensemble and trajectory thermodynamics: A brief introduction C. Van den Broeck a,∗, M. Esposito b a Hasselt University, B-3590 Diepenbeek, Belgium b Complex Systems and Statistical Mechanics, University of Luxembourg, L-1511 Luxembourg, Luxembourg h i g h l i g h t s • We review stochastic thermodynamics at the ensemble level. • We formulate first and second laws at the trajectory level. • The stochastic entropy production is the log-ratio of trajectory probabilities. • We establish the relation between the stochastic grand potential, work and entropy production. • We derive detailed and integral fluctuation theorems. article info a b s t r a c t Article history: We revisit stochastic thermodynamics for a system with discrete energy states in contact Available online 24 April 2014 with a heat and particle reservoir. ' 2014 Elsevier B.V. All rights reserved. Keywords: Stochastic Thermodynamics Entropy Fluctuation theorem Markov process 1. Introduction Over the last few years, it has become clear that one can extend thermodynamics, which is traditionally confined to the description of equilibrium states of macroscopic systems or to the transition between such states, to cover the nonequilib- rium dynamics of small scale systems. This extension has been carried out at several levels of description including systems described by discrete and continuous Markovian and non-Markovian stochastic dynamics, by classical Hamiltonian and quantum Hamiltonian dynamics and by thermostatted systems. These developments can be seen, on one side, as extend- ing the work of Onsager and Prigogine [1–4] by including microscopic dynamical properties into the far from equilibrium irreversible realm.
    [Show full text]
  • Spontaneous Symmetry Breaking and Higgs Mode: Comparing Gross-Pitaevskii and Nonlinear Klein-Gordon Equations
    S S symmetry Article Spontaneous Symmetry Breaking and Higgs Mode: Comparing Gross-Pitaevskii and Nonlinear Klein-Gordon Equations Marco Faccioli 1 and Luca Salasnich 1,2,* ID 1 Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, Via Marzolo 8, 35131 Padova, Italy; [email protected] 2 Istituto Nazionale di Ottica (INO) del Consiglio Nazionale delle Ricerche (CNR), Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy * Correspondence: [email protected] Received: 3 March 2018; Accepted: 19 March 2018; Published: 23 March 2018 Abstract: We discuss the mechanism of spontaneous symmetry breaking and the elementary excitations for a weakly-interacting Bose gas at a finite temperature. We consider both the non-relativistic case, described by the Gross-Pitaevskii equation, and the relativistic one, described by the cubic nonlinear Klein-Gordon equation. We analyze similarities and differences in the two equations and, in particular, in the phase and amplitude modes (i.e., Goldstone and Higgs modes) of the bosonic matter field. We show that the coupling between phase and amplitude modes gives rise to a single gapless Bogoliubov spectrum in the non-relativistic case. Instead, in the relativistic case the spectrum has two branches: one is gapless and the other is gapped. In the non-relativistic limit we find that the relativistic spectrum reduces to the Bogoliubov one. Finally, as an application of the above analysis, we consider the Bose-Hubbard model close to the superfluid-Mott quantum phase transition and we investigate the elementary excitations of its effective action, which contains both non-relativistic and relativistic terms.
    [Show full text]
  • Physics 451 - Statistical Mechanics II - Course Notes
    Physics 451 - Statistical Mechanics II - Course Notes David L. Feder January 8, 2013 Contents 1 Boltzmann Statistics (aka The Canonical Ensemble) 3 1.1 Example: Harmonic Oscillator (1D) . 3 1.2 Example: Harmonic Oscillator (3D) . 4 1.3 Example: The rotor . 5 1.4 The Equipartition Theorem (reprise) . 6 1.4.1 Density of States . 8 1.5 The Maxwell Speed Distribution . 10 1.5.1 Interlude on Averages . 12 1.5.2 Molecular Beams . 12 2 Virial Theorem and the Grand Canonical Ensemble 14 2.1 Virial Theorem . 14 2.1.1 Example: ideal gas . 15 2.1.2 Example: Average temperature of the sun . 15 2.2 Chemical Potential . 16 2.2.1 Free energies revisited . 17 2.2.2 Example: Pauli Paramagnet . 18 2.3 Grand Partition Function . 19 2.3.1 Examples . 20 2.4 Grand Potential . 21 3 Quantum Counting 22 3.1 Gibbs' Paradox . 22 3.2 Chemical Potential Again . 24 3.3 Arranging Indistinguishable Particles . 25 3.3.1 Bosons . 25 3.3.2 Fermions . 26 3.3.3 Anyons! . 28 3.4 Emergence of Classical Statistics . 29 4 Quantum Statistics 32 4.1 Bose and Fermi Distributions . 32 4.1.1 Fermions . 33 4.1.2 Bosons . 35 4.1.3 Entropy . 37 4.2 Quantum-Classical Transition . 39 4.3 Entropy and Equations of State . 40 1 PHYS 451 - Statistical Mechanics II - Course Notes 2 5 Fermions 43 5.1 3D Box at zero temperature . 43 5.2 3D Box at low temperature . 44 5.3 3D isotropic harmonic trap . 46 5.3.1 Density of States .
    [Show full text]