Regularized Boltzmann entropy determines macroscopic adiabatic accessibility Hiroyasu Tajima1 and Eyuri Wakakuwa2 1Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198 Japan 2Graduate School of Information Systems, The University of Electro-Communications, Chofu, Japan How the thermodynamic entropy STD is related to the Boltzmann entropy SB has been one of the central issues since the beginning of statistical mechanics. Today, it is believed that the thermodynamic entropy STD is equal to a function S˜B that is defined by regularizing the Boltzman entropy in order to ensure extensivity. However, it is not known whether S˜B completely determines the possibility of a macroscopic adiabatic transformation in the same way as STD does. In this paper, by formulating the possibility of a macroscopic adiabatic transformations in terms of “coarse- graining” of quantum operations, we prove that S˜B provides a necessary and sufficient condition for the possibility of a macroscopic adiabatic transformation. I. INTRODUCTION U, V and N [6–13]. However, the the backward im- plication of (3) is still left unproven [3]. Recently, in Thermodynamics [1, 3, 4] is one of the most successful the field of quantum information theory [15–24], it has phenomenologies in physics, and it has a huge application been succeeded in deriving detailed thermodynamic rela- from chemical reactions [1] to black holes [2]. The ther- tions, which characterize possibility and impossibility of modynamic entropy ST D plays a central role in thermo- quantum state transformations by a set of restricted op- dynamics. It is a function of macroscopic variables such erations. However, these conditions for the possibility of as U, V and N, denoting the internal energy, the volume state transformations are represented not only by macro- and the number of particles, respectively. As stated by scopic parameters, but also by microscopic parameters. the second law of thermodynamics, the thermodynamical In this sence, their results are interpreted as the “micro- entropy completely determines the macroscopic adiabatic scopic accessibility.” This is in contrast to (3), which is accessibility, [4] i.e., represented only by macroscopic parameters. In this paper, we propose a coarse-graining approach (U,V,N) (U ′, V ′,N ′) to try the backward implication in (3), and show that S˜ ≺ad B ST D[U,V,N] ST D[U ′, V ′,N ′]. (1) provides a necessary and sufficient condition for the pos- ⇔ ≤ sibility of a macroscopic adiabatic transformation, i.e., where (U,V,N) ad (U ′, V ′,N ′) means “an adiabatic the macroscopic adiabatic accessibility. Our results hold transformation from≺ a state (U,V,N) to another state by only assuming that the limit (2) exists and is a con- (U ′, V ′,N ′) is possible”. vex and strictly increasing function for U, V and N. Our How the thermodynamic entropy ST D is related to the resutls do not need stronger assumptions, e.g, the i.i.d. Boltzmann entropy SB has been one of the central issues feature. since the beginning of statistical mechanics. The Boltz- This paper is organized as follows. In the section II, mann entropy SB is not equal to thermodynamic entropy we review basic and well-known concepts of thermody- ST D in general. For example, SB is not extensive in gen- namics and statistical mechanics. All contents in this eral, while ST D is extensive. Today, it is believed that section are well-known. In the section III, we define the ST D is equal to the regularized Boltzmann entropy S˜B, macroscopic adiabatic accessibility, based on our coarse- which is defined in terms of SB as follows and is extensive graining method. In the section IV, we give the main by definition [5]: results of the present paper. Finally, in the section V, arXiv:1601.00487v2 [quant-ph] 2 Oct 2016 we prove the main results. Several lemmas used in the SB[UX,VX,NX] section V are proven in Appendix. S˜B[U,V,N] := lim . (2) X X →∞ However, it is not known whether S˜B completely deter- mines the macroscopic adiabatic accessibility in the same II. PRELIMINARIES way as ST D. That is, it is not known whether the follow- ing statement holds: In this section, we review basic concepts of thermo- dynamics and statistical mechanics. All contents in this (U,V,N) ad (U ′, V ′,N ′) section are well-known. See e.g. [3–6] for more details. ≺ S˜B[U,V,N] S˜B[U ′, V ′,N ′]. (3) In thermodynamics, an equilibrium state is represented ⇔ ≤ by values of a set of macroscopic physical quantities such In the field of statistical mechanics, the forward impli- as (U,V,N), where U is the internal energy, V is the cation of (3) has been proven for certain formulations of volume of the system, and N is the number of particles. “adiabatic operations,” by only assuming that the limit In this paper, we consider cases where all these physical (2) exists and is a convex and increasing function for quantities are extensive, the set includes the internal en- 2 (X) ergy. We represent such a set of physical quantities in with πˆ X . Let ∆ denote the set of arrays δX { ~a,δX } ∈X { } terms of vectors as ~a := (a0,a1, ..., aL), where L is a nat- such that limX δX = 0. Then ~a has a one-to-one ural number, e.g., L = 2 in the case of (U,V,N). Here, correspondence→∞ with the following set of arrays of micro- for the simplicity of writing, we express the internal en- canonical state: ergy U as a . Since a macroscopic equilibrium state is 0 (X) uniquely determined by values of macroscopic physical πˆ X δX X ∆ . (7) {{ ~a,δX } ∈X |{ } ∈X ∈ } quantities, we also represent an equilibrium state by ~a. The thermodynamical entropy of an equilibrium state is Since δX X ∆ represents the range of “negligible { } ∈X ∈ uniquely determined as a function of ~a, which we denote fluctuation” of macroscopic physical quantities, any se- by ST D[~a]. quence in the set (7) can be regarded as describing states The second law of thermodynamics is one of the most that are “macroscopically the same.” imporant assumptions in thermodynamics. It has sev- The regularized Boltzmann entropy is defined as eral equivalent formulations such as the principle of max- 1 (X) imum work, Clausius inequality and the law of entropy S˜B[~a] := lim log D ↓. (8) X X ~a increase. The law of entropy increase determines the adi- →∞ (X) (X) abatic accessibility, i.e., whether a macroscopic state can ↓ ↓ where D~a is the dimension of the Hilbert space ~a be transformed to another by an adiabatic operation. It defined by H is stated as follows: (X) (X) [l] [l] ~a := span ψ λ Xa , ~a ad ~a′ ST D[~a] ST D[~a′], (4) H n| i ∈ H ∃ ≤ ≺ ⇔ ≤ s.t. A(X)[l] ψ = λ[l] ψ for 0 l L , (9) where ~a ad ~a′ means “an adiabatic transformation | i | i ≤ ≤ o ~a ~a is≺ possible”. ad With concrete calculations, it has been shown that there →Let us introduce the statistical mechanical counterpart exists the limit S˜ as a convex and incresing function for of the thermodynamic equilibrium ~a. Since we are con- B each element of ~a, in many physical systems, e.g., gases cerning a macroscopic limit, we describe a physical sys- of particles with natural potentials including the van der tem by a Hilbert space (X) depending on a scaling pa- Waars potential [5]. rameter X. The macroscopicH limit is defined as the limit of X . We assume that X takes values in a set N→ ∞ R+ = or = . For each X and l = 0, ,L, III. FORMULATION OF MACROSCOPIC weX denote theX set of the Hermitian∈ operators X on ···(X) as H ADIABATIC ACCESSIBILITY IN TERMS OF A~(X) := (H(X), A(X),[1], ..., A(X),[L]). Then, the micro- COARSE-GRAINING canonical state corresponding to an equilibrium state ~a is defined by In this section, we introduce a general method to for- mulate possibility of a macroscopic state transforma- πˆ(X) := Πˆ (X) /D(X) , (5) ~a,δX ~a,δX ~a,δX tion, by “coarse-graining” possibility of microscopic state transformations. We then formulate the macroscopic adi- where Πˆ (X) and D(X) are the projection and the di- abatic accessibility, based on a coarse-graining of pos- ~a,δX ~a,δX mension of the following (X) , which is a subspace of of sibility of a microscopic state transformation by unital H~a,δX operations. (X): H Our formulation is based on the following idea: (X) := span ψ (X) λ[l] [X(a[l] δ ), (X) ~a,δX X Basic Idea 1 Suppose a microcanonical state π is H n| i ∈ H ∃ ∈ − ~a,δX [l] (X)[l] [l] transformed by a quantum operation X to another mi- X(a + δX )) s.t. A ψ = λ ψ for 0 l L . (X) E | i | i ≤ ≤ crocanonical state π ′ ′ . From a macroscopic point of o ~a ,δX (6) view, we observe that an equilibrium state ~a is trans- formed to another equilibrium state ~a , for any δ and The parameter δ is a positive function of X, which rep- ′ X X δ within the range of “macroscopically negligible fluc- resents the negligible fluctuation of macroscopic quanti- X′ tuations”. Therefore, we could say that an equilibrium ties. Since we are normalizing macroscopic observables state ~a can be transformed to another equilibrium state as (6), it is natural to assume that limX δX = 0. →∞ ~a′ if, for any macroscopically negligible δX and a δX′ , a To describe behavior of a system in the macroscopic (X) (X) state π can be transformed to π ′ ′ .