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arXiv:1005.3464

Microscopic expression of the second law of thermodynamics

Takaaki Monnai

Published May 19, 2010• 0 citations
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Abstract

The microscopic derivation of the second law for macroscopic system is given under the phenomenological assumption that both the initial and final states are described by mutually different canonical ensembles. In particular, it is also shown that the entropy difference between the initial and final states is composed of two positive components. One of the components is expressed as the relative entropy between the initial and final states, while the other is positive due to a dynamical stability called passivity of the canonical ensemble.

DOI
10.48550/arxiv.1005.3464

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The microscopic derivation of the second law for macroscopic system is given under the phenomenological assumption that both the initial and final states are described by mutually different canonical ensembles. In particular, it is also shown that the entropy difference between the initial and final states is composed of two positive components. One of the components is expressed as the relative entropy between the initial and final states, while the other is positive due to a dynamical stability called passivity of the canonical ensemble.
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We construct a macroscopic thermodynamical theory that satisfies the second law of thermodynamics, based on quantum mechanics. More concretely, our macroscopic theory satisfies the following two features: (1) A macroscopic equilibrium state is expressed as a set of macroscopic variables a⃗=(a1,⋯ ,aL)\vec{a}=(a_1,\cdots,a_L)a=(a1​,⋯,aL​), which represents coarse-grained values of a set of physical quantities, and (2) An adiabatic transformation from an equilibrium state a⃗\vec{a}a to another state a⃗′\vec{a}'a′ is possible if and only if sa⃗≤sa⃗′s_{\vec{a}}\le s_{\vec{a}'}sa​≤sa′​ holds. Here, sa⃗s_{\vec{a}}sa​ is a real-valued function corresponding to the thermodynamical entropy, which depends only on macroscopic variables a⃗\vec{a}a. We show that such a macroscopic thermodynamical theory can be consistently constructed if the von-Neumann entropy of a generalized microcanonical state satisfies a generalized central limit theorem. We also show that even when the von-Neumann entropies do not satisfy the condition, a macroscopic thermodynamic-like theory with a similar structure exists, which could be at the foundations of nonequilibrium thermodynamics. Finally, we make a replica of (U,V,N)(U,V,N)(U,V,N) expression of thermodynamics satisfying (1) and (2), using a toy model of free particles in a box.
Based on quantum statistical mechanics and microscopic quantum dynamics, we prove Planck's and Kelvin's principles for macroscopic systems in a general and realistic setting. We consider a hybrid quantum system that consists of the thermodynamic system, which is initially in thermal equilibrium, and the "apparatus" which operates on the former, and assume that the whole system evolves autonomously. This provides a satisfactory derivation of the second law for macroscopic systems.
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Deriving the laws of thermodynamics from an underlying microscopic picture is a central quest of statistical mechanics. The present article focuses on the derivation of the first and second law of thermodynamics for closed and open quantum systems, where such foundational questions in addition become practically relevant for emergent nanotechnologies. Recent progress is presented in a self-contained way, with an emphasis on general ideas and useful tools instead of particular applications. We identify limits and conceptual shortcomings of previous approaches. Then, we propose a novel, unifying perspective, which starts from a microscopic definition of nonequilibrium thermodynamic entropy. The change of this entropy is identified with the entropy production, which satisfies a fluctuation theorem for a large class of initial states. Moreover, this entropy production can be naturally separated into a quantum and a classical component. Within our framework, we introduce the notions of recoverable work and remaining heat. In the case of a weakly perturbed ideal thermal bath this approach is in quantitative agreement with previous ones, thus ensuring their thermodynamic consistency.