Portland State University PDXScholar Physics Faculty Publications and Presentations Physics 8-8-2018 Supercanonical Probability Distributions John D. Ramshaw Portland State University, [email protected] Follow this and additional works at: https://pdxscholar.library.pdx.edu/phy_fac Part of the Physics Commons Let us know how access to this document benefits ou.y Citation Details Ramshaw, J. D. (2018). Supercanonical probability distributions. Physical Review E, 98(2), 020103. This Article is brought to you for free and open access. It has been accepted for inclusion in Physics Faculty Publications and Presentations by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. PHYSICAL REVIEW E 98, 020103(R) (2018) Rapid Communications Supercanonical probability distributions John D. Ramshaw Department of Physics, Portland State University, Portland, Oregon 97207, USA (Received 19 February 2018; published 8 August 2018) The canonical probability distribution describes a system in thermal equilibrium with an infinite heat bath. When the bath is finite the distribution is modified. These modifications can be derived by truncating a Taylor-series expansion of the entropy of the heat bath, but their form depends on the expansion parameter chosen. We consider two such expansions, which yield supercanonical (i.e., higher-order canonical) distributions of exponential and power-law form. The latter is identical in form to the “Tsallis distribution,” which is therefore a valid asymptotic approximation for an arbitrary finite heat bath, but bears no intrinsic relation to Tsallis entropy. DOI: 10.1103/PhysRevE.98.020103 −1 The probability that a system in thermal equilibrium with an and Cb(E) ≡ (∂Tb/∂E) . Note that entropy is dimensionless infinite heat bath occupies microstate i is given by the canonical when temperature is expressed in energy units, which is for- distribution mally equivalent to replacing Boltzmann’s constant by unity. −1 pi = Z exp{−βEi }, (1) If the heat bath is a proper thermodynamic system, then as it becomes infinitely large Tb(E) will approach a finite limiting where Ei is the energy of microstate i, β = 1/Tb, Tb is the temperature of the heat bath in energy units, and Z(β) ≡ value, while E itself and Cb(E) become infinite. The latter i divergence reflects the fact that it requires infinite energy to exp{−βEi } is the canonical partition function. Equation (1) is widely regarded as the single most important formula produce a finite change in the temperature of an infinite system. in statistical physics. Our purpose here is to explore the The standard derivation of the canonical distribution from corrections or modifications to it that arise when the heat bath Eq. (2) is based on the assumption that the heat bath is much larger than the system, so that Ei E0 and a Taylor series is finite. Such corrections represent an important aspect of the − = − thermodynamics and statistical mechanics of small systems expansion of Sb(E0 Ei ) log Wb(E0 Ei )inpowersof [1–3], which are experiencing a lively renaissance (e.g., [4–8]) Ei can reasonably be presumed to converge rapidly [13]. stimulated in part by rapid advances in nanotechnology. Much Equation (1) is then an immediate consequence of truncating of the literature on finite heat baths has been narrowly focused this expansion at the linear term [13–15]. However, this on the observation that simple ideal heat baths imply power-law procedure further implies that Eq. (1) is not exact but is in probability distributions of the same form as those obtained by principle modified by the higher-order terms in the expansion, extremizing the Tsallis entropy [9–11]. This correspondence at least for finite values of E0. It is instructive and enlightening has often been interpreted as evidence in support of the Tsallis to examine the form of those modifications when the quadratic E2 entropy, but that inference is unjustified [12]. term of order i [13–16] is retained. Thus we proceed to The probability distribution for a system in thermal equilib- explore the implications of the approximation rium with a finite heat bath is fundamentally given by [13–15] − = − − 2 Sb(E0 Ei ) Sb(E0 ) β1Ei β2Ei , (5) W (E − E ) p = b 0 i , i − (2) where j Wb(E0 Ej ) where E is the total energy of the system and bath together, and ∂Sb 1 0 β ≡ = , (6) W (E) is the number of bath states with energies in the interval 1 0 b ∂E 0 Tb (E,E+E), where E is a macroscopically small but finite 1 ∂2S β energy tolerance. Equation (2) is based on the hypothesis of β ≡− b = 1 , (7) 2 2 0 0 equal aprioriprobabilities, according to which those bath 2 ∂E 0 2Cb Tb states are presumed to be equally probable, and the further = 0 ≡ subscript zero denotes evaluation at E E0, Tb Tb(E0 ), presumption that the heat bath is sufficiently large and complex 0 ≡ =∼ and Cb Cb(E0 ). Combining Eqs. (2), (3), and (5), we obtain that Wb(E) ωb(E) E, where ωb(E) is a differentiable continuous approximation to the true discontinuous density = −1 − − 2 pi Z2 exp β1Ei β2Ei , (8) of bath states [15]. If the heat bath were isolated with energy E, its entropy S , temperature T , and heat capacity C would where b b b be given by ≡ − − 2 Z2(β1,β2 ) exp β1Ei β2Ei . (9) i Sb(E) = log Wb(E), (3) It follows from Eq. (7) that β2 vanishes for an infinite heat bath, 1 ∂Sb = , (4) whereupon Z2(β1,β2 ) reduces to Z(β) and Eq. (8) reduces to Tb(E) ∂E Eq. (1). If higher-order terms had been retained in Eq. (5), 2470-0045/2018/98(2)/020103(3) 020103-1 ©2018 American Physical Society JOHN D. RAMSHAW PHYSICAL REVIEW E 98, 020103(R) (2018) they too would clearly vanish in the same limit. The canonical Wb(E) in finite ideal systems, such as ideal gases or harmonic distribution therefore becomes exact when the heat bath is solids, is typically proportional to Eα, where the exponent α is infinite, but not otherwise. Its universality derives from the fact on the order of the number of particles or degrees of freedom that the only bath parameter it depends upon is the temperature. of the system [9,13–15]. This suggests that faster convergence According to Eq. (8), the probabilities pi are unequal, so the might be obtained by regarding Sb(E) = log Wb(E)asa entropy S of the system must be evaluated as the Boltzmann- function of log E rather than E itself. We therefore proceed Gibbs-Shannon (BGS) entropy to approximate Sb(E) − Sb(E0 ) by a truncated Taylor series expansion in powers of log E − log E . Since E will remain =− 0 0 S pi log pi , (10) large but finite, the lowest-order correction to the canonical i distribution is obtained by truncating the expansion at the linear which contrary to a common misconception is not uniquely or term. Thus we write inherently associated with a canonical (or any other) probabil- S (E) = S (E ) + α(log E − log E ) ity distribution [15]. Combining Eqs. (8) and (10), we obtain b b 0 0 α = + E = + 2 + Sb(E0 ) log , (17) S β1E β2E log Z2, (11) E 0 ≡ ≡ = where Q Qi i pi Qi . When β2 0, Eq. (11) reduces where to the standard canonical expression for the entropy S.The ∂S ∂S E quadratic term in Eq. (5) therefore introduces an additional ≡ b = b = = 0 α E0 E0β1 0 (18) 2 ∂ log E ∂E T term β2E into the entropy of the system. It follows from 0 0 b Eq. (9) that and use has been made of Eq. (6). Combining Eqs. (2), (3), (17), and (18), we obtain ∂ log Z2 =−E, (12) α ∂β − β1Ei 1 p = Z 1 1 − , (19) ∂ log Z i α 2 =−E2, (13) ∂β2 where which combine with Eq. (11) to imply that α β1Ej Z(α, β1) ≡ 1 − . (20) = + 2 α dS β1 dE β2 dE . (14) j Note, however, that E2 is not an independent variable, because Aside from minor notational differences, Eq. (19)isof 2 2 E and E are both functions of E0,sodE and dE are both precisely the same form as the so-called Tsallis distribu- simply proportional to dE0 and therefore to each other. Thus tion previously obtained by extremizing the Tsallis entropy dS is simply proportional to dE, so that the second law of [17], and subsequently legitimized by the observation that it thermodynamics remains valid in the form dE = TdS, which describes a system in equilibrium with an ideal finite heat defines and determines the temperature T of the system. Thus bath for which Wb(E) has the same power-law form [9]. The present derivation shows that this distribution is actually 1 = dS = + = 1 + R much more general: it is not restricted to ideal heat baths β1 Rβ2 0 1 0 0 , (15) T dE Tb 2Cb Tb but rather represents a first-order asymptotic approximation to S (E) for an arbitrary finite heat bath with an arbitrary where b quasicontinuous density of states. This derivation also provides dE2 dE2/dE exact expressions for the parameters α and β1 in terms of R ≡ = 0 . (16) the caloric equation of state Tb(E) of the heat bath and the dE dE/dE0 total energy E0. Moreover, the systematic expansion procedure The parameters β1 and β2 are functions of E0 via Eqs. (6) on which this approximation is based could in principle be and (7), so T can be evaluated by substituting Eq.