Statistical mechanical derivation of the second law of thermodynamics

Hal Tasaki

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Statistical mechanical derivation of the second law of thermodynamics

Hal Tasaki

Statistical Mechanics · Submitted 14 Sep 2000

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The authors' main claim Therefore the second law is a logical necessity once we accept equilibrium statistical mechanics.

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In a macroscopic (quantum or classical) Hamiltonian system, we prove the second law of thermodynamics in the forms of the minimum work principle and the law of entropy increase, under the assumption that the initial state is described by a general equilibrium distribution. Therefore the second law is a logical necessity once we accept equilibrium statistical mechanics.

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s0	In a macroscopic (quantum or classical) Hamiltonian system, we prove the second law of thermodynamics in the forms of the minimum work principle and the law of entropy increase, under the assumption that the initial state is described by a general equilibrium distribution.	contribution 0.55	0.93	0.60	asserted 1.00
s1	Therefore the second law is a logical necessity once we accept equilibrium statistical mechanics.	interpretation 1.00	0.50	0.81	asserted 1.00

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Original abstract

In a macroscopic (quantum or classical) Hamiltonian system, we prove the second law of thermodynamics in the forms of the minimum work principle and the law of entropy increase, under the assumption that the initial state is described by a general equilibrium distribution. Therefore the second law is a logical necessity once we accept equilibrium statistical mechanics.

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References

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Thermodynamic Speed Limits in Isolated Quantum Systems
cond-mat.stat-mech 2026-09 accept novelty 7.0 of 10

Entropy production in isolated quantum systems is bounded from below by the square of the Vu-Saito quantum Wasserstein distance, yielding thermodynamic speed limits for coarse-grained state transformations.
Jarzynski Relations for Quantum Systems and Some Applications
cond-mat.stat-mech 2000-09 conditional novelty 4.0 of 10

Quantum Jarzynski equalities yield a statistical-mechanical derivation of the entropy increase law for compound systems and a nonnegative heat-flow bound between two quantum systems at different temperatures.

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15 arXiv papers cite this in 32 sentences

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Builds on it 1 paper

Specifically, we present generalised versions of the Tasaki-Crooks relation (TCR) [ref] and the quantum Jarzynski equality (QJE) [ref] suitable to describe fluctuations in systems with an arbitrary, and possibly variable, number of charges.

Revealing missing charges with generalised quantum fluctuation relations · Introduction · 1.00

Compares with it 2 papers

[ref] already proposed the quantum entropy operator in the form $𝒮 = ln𝒩$, but the connection with the Generalized Helmholtz Theorem (which has been introduced later [ref] ) was not made, neither the importance of the volume entropy as a good mechanical analogue of thermodynamic entropy for possibly low dimensional systems was recognized.

Mechanical Proof of the Second Law of Thermodynamics Based on Volume Entropy · Thomson's formulation and the Minimal Work Principle · 0.88
For processes identical to the ones analyzed in Secs. [ref] and [ref] of the present paper – that is, canonically sampled initial conditions[ In fact, Tasaki and Campisi require only the weaker assumption that initial conditions be sampled from a distribution that is a monotonically non-increasing function of $H_A$. ] followed by Hamiltonian or unitary evolution – Tasaki [ref] and Campisi [ref] have derived inequalities that are expressed in our notation as follows (see Eq. 9 of Ref. [ref] and Eqs. 16 and 36 of Ref. [ref] ):

Fluctuation relations and strong inequalities for thermally isolated systems · Discussion · 0.92

Cites it as support 1 paper

Very recently it has been also proved that non-adiabatic transformations occurring in isolated systems which are initially in a state of thermal equilibrium always result in an increase of the expectation value of the volume entropy [ref] .

Microcanonical phase transitions in small systems · Why volume entropy · 0.96

Cites it as background 3 papers

We start by briefly reviewing the main results in Ref. [ref] , in particular the generalised versions of the quantum Jarzynski [ref] and Tasaki-Crooks [ref] relations.

Fluctuations of work in realistic equilibrium states of quantum systems with conserved quantities · Review of generalized quantum fluctuation relations · 1.00
Regarding the law of increasing entropy, it has been discussed based on certain forms of entropy, such as extensions of the Boltzmann entropy [ref] , diagonal entropy [ref] , and observable entropy [ref] .

Second law of thermodynamics in closed quantum many-body systems · Introduction · 0.98
This question brings us to the foundation of the second law of thermodynamics [ref] .

Work Extraction from a Single Energy Eigenstate · Introduction · 0.98

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