Pith. sign in

REVIEW 4 major objections 4 minor 47 references

A Rigorous Foundation for Stochastic Thermodynamics via the Microcanonical Ensemble

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that a small system strongly coupled to a large, fast bath has Markovian reduced dynamics obeying $P_F(Q,x_t|x_0;\beta)/P_B(-Q,x_0^*|x_t^*;\beta)=e^{-\beta Q}$, with heat defined as minus the bath-energy change.

desk verdict A clever microcanonical derivation of local detailed balance that mostly delivers, but the Markovianity proof rests on an explicit assumption the abstract overstates. read the letter →

arxiv 2506.23604 v2 pith:RL4KS54Q submitted 2025-06-30 cond-mat.stat-mech cond-mat.soft

Xiangjun Xing (Wilczek Quantum Center , School of Physics , Astronomy , Shanghai Jiao Tong University , Shanghai 200240 , China) This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.soft PACS 05.70.Ln05.40.-a05.20.-y
keywords stochasticthermodynamicsmicrocanonicalensemblelocaldetailedbalanceMarkovianityHamiltonianofmeanforceadiabaticinvariancefluctuationtheoremsstrongcoupling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Stochastic thermodynamics usually assumes weak coupling and works with path probabilities, which are subtle to define when noise is multiplicative or coupling is strong. This paper replaces that setting with an isolated Hamiltonian system split into a slow system $X$ and a fast, large bath $Y$, assuming only that the bath equilibrates much faster than the system and that the coarse-grained joint dynamics is Markovian. It proves the reduced $X$-dynamics is then Markovian and satisfies local detailed balance at the level of transition probabilities, $P_F(Q,x_t|x_0;\beta)/P_B(-Q,x_0^*|x_t^*;\beta)=e^{-\beta Q}=e^{\Delta S_Y}$, where $Q$ is the heat released by the bath and $\Delta S_Y$ is its Boltzmann entropy change. Heat and bath entropy are defined directly from the microcanonical bath rather than inferred from system observables, so the relations hold at arbitrary system-bath coupling. If correct, the paper supplies a model-independent foundation from which fluctuation theorems follow without path integration.

What carries the argument

The load-bearing object is a decomposition of the total Hamiltonian, $H_{\mathrm{tot}}(x,y,\lambda)=H_X(x;\lambda)+H_Y(y;x,\lambda)$, chosen so that the bath microcanonical partition function $\Omega_Y(E_Y;x,\lambda)$ does not depend on the system state $x$ or the control parameter $\lambda$. Starting from any split, the paper constructs a subextensive correction $\chi$ from the entropy change of the bath under variations of $x$ and $\lambda$, and shifts the bath Hamiltonian by $T\chi$; this makes the bath energy $E_Y$ an adiabatic invariant and forces $H_X$ to be the Hamiltonian of mean force. The decomposition ensures $dE_Y=T\,dS_Y=-d\bar Q$, so heat has a definite microscopic meaning, and it lets the generalized detailed balance relation for the coarse-grained joint dynamics be integrated over bath variables into the transition-level local detailed balance ratio. The assumed time-scale separation is what keeps the bath in instantaneous microcanonical equilibrium conditioned on the system state and control parameter.

What would settle it

Simulate a Hamiltonian model with a small strongly coupled system and a large bath under a slow driving protocol; coarse-grain phase space into cells. Compute $P_F(Q,x_t|x_0;\beta)$ by counting transitions among cells and compare with $P_B(-Q,x_0^*|x_t^*;\beta)$, with $Q$ taken as minus the bath-energy change. A systematic deviation from $e^{-\beta Q}$, or a failure of the Chapman-Kolmogorov equality for coarse-grained times longer than the bath mixing time, would refute the central claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that time reversal plus time-scale separation is enough to put stochastic thermodynamics on a microcanonical footing. For a total Hamiltonian split as $H_{\mathrm{tot}}(x,y,\lambda)=H_X(x;\lambda)+H_Y(y;x,\lambda)$, with $H_Y$ chosen so that the bath microcanonical partition function is independent of $x$ and $\lambda$, the bath energy is an adiabatic invariant and $H_X$ is the Hamiltonian of mean force. Under these conditions the coarse-grained forward and backward transition probabilities of the joint system satisfy generalized detailed balance, and integrating over the fast bath variables yields the local detailed balance ratio with exponent $-\beta Q$. This is equivalent to the conventional trajectory-level local detailed balance when path probabilities are defined by discretization, but it is free of the multiplicative-noise ambiguities in path measures. From the transition-level relation the paper derives a master fluctuation theorem for the joint statistics of work, heat, and system states, then marginalizes it to obtain the Crooks work fluctuation theorem and additional joint fluctuation relations, several of which it identifies as not previously discussed.

Load-bearing premise

The paper explicitly assumes, rather than proves, that the coarse-grained joint dynamics of system and bath is Markovian, meaning memory-free; if the bath does not decorrelate fast enough or memory persists beyond the bath mixing time, the Chapman-Kolmogorov step and the derivation of local detailed balance from it fail.

Editorial extensions

If this is right

  • Local detailed balance holds for arbitrary strength and tunable coupling between system and bath, so fluctuation theorems do not require the usual weak-coupling limit.
  • Work and heat can be computed from the system trajectory alone: work from the control parameter and non-conservative force, heat from the change in the system Hamiltonian minus work.
  • The transition-probability form of LDB avoids defining path measures, removing a known source of ambiguity for multiplicative-noise processes.
  • A single master fluctuation theorem for work, heat, and initial and final system states yields the Crooks work fluctuation theorem and joint fluctuation relations involving system states.
  • The construction extends immediately to multiple baths, with the LDB ratio carrying $e^{-\sum_k \beta_k Q_k}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct numerical test is possible: simulate a strongly coupled Hamiltonian system with a slow protocol, compute $Q$ from the bath energy change, coarse-grain the dynamics, and check whether the measured forward/backward transition ratio equals $e^{-\beta Q}$ even where weak-coupling LDB fails.
  • The assumed Markovianity of the coarse-grained joint dynamics is separable from the LDB claim; if that assumption fails, the Chapman-Kolmogorov route breaks down, yet the ratio itself could still hold in specific non-Markovian models, so the two layers should be tested independently.
  • The shift in the bath Hamiltonian by $T\chi$ acts like a gauge freedom: different choices of the split leave bulk properties unchanged, which suggests an organizing principle for choosing thermodynamic potentials in strong-coupling theories beyond the specific construction given here.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a microcanonical derivation of Markovianity and local detailed balance (LDB) for a small system strongly coupled to a large Hamiltonian bath. It constructs a decomposition of the total Hamiltonian into an 'adiabatically invariant' bath Hamiltonian and a system Hamiltonian of mean force, defines heat as the negative change of bath energy, and, assuming time-reversal symmetry, time-scale separation, and an explicitly assumed Markovian coarse-grained joint dynamics, derives a generalized detailed balance relation (GDB). From GDB it derives a transition-probability-level LDB relation and then a master fluctuation theorem, together with Crooks and related fluctuation theorems. The paper explicitly acknowledges in the Introduction that Markovianity of the joint dynamics is assumed, not derived from microscopic principles, but the Abstract and concluding remarks present the Markovianity result as if it followed from time-scale separation and time-reversal symmetry alone.

Significance. If the stated assumptions are granted, the paper gives a clean and largely model-independent route from a microcanonical bath description to the LDB ratio and to fluctuation theorems, while avoiding path-probability subtleties and remaining valid at strong system-bath coupling. The construction of the Hamiltonian of mean force via a subextensive gauge-like shift is interesting and connects naturally with earlier strong-coupling thermodynamics. A genuine strength is that the logical chain from GDB to the LDB ratio (Section VI) and to the master fluctuation theorem (Section VII) is explicit and checkable, and the author is honest about the unproven joint-Markovianity premise. The main weakness is the gap between the Abstract's claim of a rigorous, model-free derivation of Markovianity and the actual conditional theorem, which requires the joint Markovianity assumption plus an additional locality/extensivity assumption in Eq. (4.9). Because the advertised contribution is a 'rigorous foundation,' this gap is load-bearing and needs to be addressed in revision.

major comments (4)
  1. [Abstract; Section IV, Eq. (4.1)] The Abstract states that 'using time-reversal symmetry and TSS, and without invoking any model details, we rigorously prove that the reduced dynamics of the system is Markovian.' This overstates what is proved. Equation (4.1) is introduced as an assumption ('We assume that ... the coarse-grained joint dynamics ... becoming Markovian'), and the Introduction's 'One important clarification' explicitly says Markovianity of the joint dynamics is assumed, not derived from microscopic principles. Equations (4.10)-(4.11) propagate that assumption to the reduced dynamics; they do not establish Markovianity from TSS alone. The theorem should be restated with its full hypothesis, and the Abstract should not claim a model-free derivation of Markovianity.
  2. [Section II.A, Eqs. (2.14)-(2.18)] The construction of the adiabatically invariant bath Hamiltonian has a self-consistency issue. In Eq. (2.15), H_Y(y;x,lambda,E_Y) depends on the energy E_Y that also serves as the constraint variable in the microcanonical density (2.2), so the energy shell is defined through H_Y(y;x,lambda,E_Y)=E_Y. When the delta function in Eq. (2.17) is evaluated, the shift in the argument of H0_Y is T*chi (an energy), so one obtains Omega0_Y(E_Y - T*chi), not Omega0_Y(E_Y - chi) as printed; the displayed expression subtracts an entropy from an energy. Furthermore, Eq. (2.18) is only a first-order expansion in the subextensive chi, so the claimed independence partial S_Y/partial x = partial S_Y/partial lambda = 0 holds only up to corrections of order chi/E_Y. Since this independence is used in Eq. (2.10), Eq. (5.12), and the LDB exponent, the approximation order should be stated explicitly and the self-consistent definition of the energy shell should be given.
  3. [Section IV, Eq. (4.9)] The reduction of the reduced transition probability's dependence on the initial bath energy E_Y0 to a dependence on the temperature beta alone is asserted rather than derived. The sentence 'A crucial observation is that the initial bath energy can influence the reduced X dynamics only through the temperature' introduces a locality/extensivity assumption that is not formalized. Because Eq. (4.9) is used in the Chapman-Kolmogorov argument, the Markovianity theorem inherits this additional physical assumption. It should be listed explicitly among the hypotheses, and the conditions under which the E_Y0 dependence drops out should be stated.
  4. [Section VI, Eqs. (6.5)-(6.8)] The advertised LDB relation e^{-beta Q} is obtained by the Taylor expansion in Eq. (6.7) and is therefore approximate, valid for |Q| << E_Y0 and negligible temperature change. The exact relation that follows from GDB and the definitions is Eq. (6.5), with exponent S_Y(E_Y0-Q)-S_Y(E_Y0), which is not generally equal to -beta Q for a finite bath. If the paper's goal is a rigorous foundation, the theorem should be stated in the exact form and the large-bath limit leading to Eq. (6.8) should be spelled out, including the uniformity of the limit or the precise smallness condition on Q.
minor comments (4)
  1. [Throughout] There are numerous typographical errors, including 'well-konwn' (Section I), 'sahall' (Section III A), 'intrdouce' (after Eq. (4.4)), 'evovles' and 'yy' (Section VII), 'noramlized' (Section VII), and 'cubersome' (Section VI). A careful proofreading pass is needed.
  2. [Section VI, Eq. (6.2)] The notation P_F(Q,x_t|x_0;beta) suppresses the initial bath energy E_Y0 that appears on the right-hand side of Eq. (6.2) and in the conditioning of the angle-bracket transition probability. Either include E_Y0 in the conditioning or state explicitly that the dependence is dropped under the extensivity assumption of Eq. (4.9).
  3. [Section VII, Eq. (7.13)] The ratio of singular densities p_F/p_B in Eq. (7.13) is taken on the support of the delta function enforcing the first law in Eq. (7.9). This should be stated explicitly, since outside that support the ratio is not well defined.
  4. [References] Reference [34] is identical to Reference [28] ('Time-slicing path-integral in curved space'); the duplicate entry should be removed or replaced with the intended citation.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: LDB and the fluctuation theorems are derived from generalized detailed balance plus explicit microcanonical/TSS inputs; the ΔS_Y = -βQ identification is definitional via the chosen H_Y decomposition, and reduced Markovianity is an admitted conditional transfer from an assumed joint-Markovianity premise.

full rationale

The paper's load-bearing derivations are not circular: the generalized detailed balance (GDB) of Sec. III (Eqs. 3.12-3.16) is derived in-paper from microscopic time-reversal symmetry, Liouville volume conservation, and Boltzmann-cell coarse-graining; the LDB ratio (Eq. 6.8) follows from GDB by multiplying the forward/backward kernels by the microcanonical bath density of states e^{S_Y} supplied by the TSS assumption (Eqs. 4.2, 4.8, 6.2, 6.4), so the ratio P_F/P_B = e^{ΔS_Y} is a genuine derived relation, and the fluctuation theorems of Sec. VII follow from the same ingredients. The second equality, ΔS_Y = -βQ (Eqs. 6.6-6.7), is definitional: β ≡ ∂S_Y/∂E_Y and Q ≡ E_{Y,0} - E_{Y,t}, so it is a Taylor identity once the paper has chosen the decomposition H_Y so that S_Y depends only on E_Y ('Our goal is therefore to find the particular decomposition... such that ∂S_Y/∂λ = ∂S_Y/∂x = 0', Sec. II). The advertised novelty (Sec. VIII: 'demonstrating that the quantity -βQ precisely corresponds to the change in the Boltzmann entropy of the bath') is therefore largely a definitional identification rather than a discovery, but it does not make the LDB ratio itself an input. The Markovianity result is explicitly conditional: the paper concedes 'The Markovianity of the joint dynamics is assumed, not derived from microscopic principles' (Introduction) and 'The exact mathematical conditions that guarantee the Markovian nature of the coarse-grained dynamics is currently not understood' (Sec. IV); the abstract's claim that TSS alone 'rigorously prove[s] that the reduced dynamics of the system is Markovian' overstates this, since Eq. (4.11) merely transfers assumed joint Markovianity to the reduced process. That is a rigor/premise concern, not circularity. Self-citations ([17], [26], [27], [32], [33], [34]) are consistency/comparison statements ('fully consistent with our earlier works', 'agrees with the result derived in Ref. [32]') and none is load-bearing in the derivation chain. Verdict: no circular reduction found; score 2 reflecting the density of non-load-bearing self-citation and the construction-driven entropy/heat identification, both disclosed in the text.

Assumptions & free parameters 1 free parameters · 8 assumptions · 1 invented entities

The central derivation rests on several physical assumptions that are acknowledged in the text: time-scale separation, Markovianity of the coarse-grained joint dynamics, time-reversal symmetry, and the subextensive and localized nature of the system-bath interaction. There are no fitted numerical parameters in the usual sense; the only hand-chosen quantity is the reference state used to define the gauge-transformed bath Hamiltonian, which cancels from physical predictions.

free parameters (1)
  • Reference state (x0, lambda0) in the Hamiltonian construction = arbitrary; chosen by hand
    Eq. (2.14) defines chi relative to arbitrary reference values x0 and lambda0. The resulting H_Y and H_X depend on this choice, but the LDB ratio, heat, and free-energy differences do not. It is a gauge degree of freedom, not a parameter fitted to data.
assumptions (8)
  • domain assumption Time-scale separation (TSS)
    The bath equilibrates microcanonically on a timescale much shorter than system and driving timescales. Invoked in the Introduction, Eq. (2.2), and Section IV.
  • domain assumption Markovianity of the coarse-grained joint (X,Y) dynamics
    Section IV assumes the coarse-grained joint dynamics satisfies Chapman-Kolmogorov for times longer than the bath mixing time; the paper explicitly says this is not derived from microscopic principles.
  • domain assumption Time-reversal symmetry of microscopic and coarse-grained dynamics
    Section III assumes H_X and H_Y are invariant under time reversal with reversed variables and protocol; used to derive GDB Eq. (3.15).
  • domain assumption Locality and subextensivity of system-bath coupling
    Changes in x and lambda alter the bath only in a localized interfacial region, so S0_Y changes subextensively and the bath temperature is independent of x and lambda. Stated via the colloid-in-fluid example in the Introduction and used in Eqs. (2.14)-(2.18).
  • domain assumption Microcanonical equal-a-priori postulate for the conditioned bath
    The bath is uniformly distributed on the energy surface H_Y=E_Y for given x and lambda, Eq. (2.2). This is taken as the definition of instantaneous equilibrium under TSS.
  • domain assumption Validity of first-order Taylor expansions of bath entropy
    The heat Q and the gauge shift chi are assumed subextensive relative to the bath energy, so S_Y(E-Q) and S0_Y(E-chi) can be expanded to first order; used in Eqs. (2.18), (6.7), and (7.2).
  • domain assumption Ergodic time-averaging of fast bath variables
    The work done by the control parameter on the bath is replaced by its microcanonical ensemble average in Eq. (5.12), assuming the fast bath explores the energy surface within the slow timescale.
  • standard math Liouville theorem and volume preservation under time reversal
    Used in Section III to obtain |UA|=|A| and |A*|=|A|, leading to GDB Eq. (3.15).
invented entities (1)
  • Adiabatically invariant bath Hamiltonian H_Y(y;x,lambda)
    purpose: Redefines the system-bath decomposition so that the bath entropy is independent of x and lambda and heat equals minus the bath energy change.
    H_Y is constructed in Eq. (2.15) from an arbitrary H0_Y plus a subextensive gauge shift T chi. It is a mathematical redefinition rather than a new physical force or particle, and it has no independent experimental handle beyond the theory's internal definitions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Rigorous Foundation for Stochastic Thermodynamics via the Microcanonical Ensemble." pith.science (2026). https://pith.science/paper/RL4KS54Q

@misc{pith2026250623604,
  author       = {Pith},
  title        = {Pith review of: A Rigorous Foundation for Stochastic Thermodynamics via the Microcanonical Ensemble},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RL4KS54Q}},
  note         = {Machine review of arXiv:2506.23604}
}
read the original abstract

We consider a small Hamiltonian system strongly interacting with a much larger Hamiltonian system (the bath), while being driven by both a time-dependent control parameter and non-conservative forces. The joint system is assumed to be thermally isolated. Under the assumption of time-scale separation (TSS)--where the bath equilibrates much faster than the system and the external driving--the bath remains in instantaneous equilibrium, described by the microcanonical ensemble conditioned on the system state and the control parameter. We identify a decomposition of the total Hamiltonian that renders the bath energy an adiabatic invariant under slow evolution. This same decomposition defines the system Hamiltonian as the Hamiltonian of mean force, and ensures that neither the system nor the control parameter does reactive work on the bath. Using time-reversal symmetry and TSS, and without invoking any model details, we rigorously prove that the reduced dynamics of the system is Markovian and satisfies a form of local detailed balance (LDB) which involves transition probabilities but not path probabilities. By working entirely within the microcanonical framework and adopting a precise decomposition of the total energy, we provide rigorous definitions of bath entropy as the Boltzmann entropy, and of heat as the negative change of the bath energy. Our approach bypasses the ambiguities associated with conventional definitions of thermodynamic variables and path probabilities, and establishes a rigorous and thermodynamically consistent foundation for stochastic thermodynamics, valid even under strong system-bath coupling.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 42 canonical work pages

  1. [1]

    We may assume that the conservative force also depends onλ, so that its functional forms aref(z, λ s) andf(z, ˜λs) in the forward and backward processes respectively

    Note that in both the forward process and the backward process, the time runs from 0 tot. We may assume that the conservative force also depends onλ, so that its functional forms aref(z, λ s) andf(z, ˜λs) in the forward and backward processes respectively. These defines the dynamic protocols of the forward and backward processes. The initial states of the...

    work page

  2. [2]

    Combining the preceding two equations, we find ˜U(U z0)∗ =z ∗ 0.(3.5) Now letAbe a set, e.g

    In the below, we shall use simplified notationsU=U(t,0) and ˜U= ˜U(t,0), as there is no danger of confusion. Combining the preceding two equations, we find ˜U(U z0)∗ =z ∗ 0.(3.5) Now letAbe a set, e.g. a Boltzmann cell, in the state space ofz. We defineU Aas the image ofAcreated byU: U A≡ {U z|z∈A}.(3.6) Basic set theory then tells usU(A∩B) =U A∩U BandU(A...

    work page

  3. [3]

    Peliti and S

    L. Peliti and S. Pigolotti,Stochastic Thermodynamics: An Introduction(Princeton University Press, 2021)

    2021

  4. [4]

    Equalities and Inequalities: Irreversibility and the Second Law of Thermody- namics at the Nanoscale,

    C. Jarzynski, “Equalities and Inequalities: Irreversibility and the Second Law of Thermody- namics at the Nanoscale,”Annu. Rev. Condens. Matter Phys.2, 329–351 (2011)

    work page 2011

  5. [5]

    Sekimoto, Stochastic energetics of Brownian motion, Phys

    K. Sekimoto, Stochastic energetics of Brownian motion, Phys. Rev. Lett.80, 3424 (1998)

    work page 1998

  6. [6]

    Stochastic thermodynamics, fluctuation theorems and molecular machines,

    U. Seifert, “Stochastic thermodynamics, fluctuation theorems and molecular machines,”Rep. Prog. Phys.75, 126001 (2012)

    work page 2012

  7. [7]

    Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theo- rem, Phys

    U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theo- rem, Phys. Rev. Lett.95, 040602 (2005)

    work page 2005

  8. [8]

    Jarzynski, Nonequilibrium equality for free energy differences, Phys

    C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett.78, 2690 (1997)

    work page 1997

Show all 47 references
  1. [9]

    G. N. Bochkov and Yu. E. Kuzovlev, General theory of thermal fluctuations in nonlinear systems, Sov. Phys. JETP45, 125 (1977). 29

    1977

  2. [10]

    G. N. Bochkov and Yu. E. Kuzovlev, Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics I, Physica A106, 443 (1981)

    1981

  3. [11]

    Crooks, G. E. (1999). Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences.Physical Review E, 60(3), 2721 (1999)

    1999

  4. [12]

    J. L. Lebowitz and H. Spohn, A Gallavotti-Cohen-type symmetry in the large deviation func- tional for stochastic dynamics, J. Stat. Phys.95, 333 (1999)

    1999

  5. [13]

    Gallavotti and E

    G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in stationary states, J. Stat. Phys. 80, 931 (1995)

    1995

  6. [14]

    Zwanzig, Ensemble method in the theory of irreversibility, J

    R. Zwanzig, Ensemble method in the theory of irreversibility, J. Chem. Phys.33, 1338 (1960)

    1960

  7. [15]

    Mori, Transport, collective motion, and Brownian motion, Prog

    H. Mori, Transport, collective motion, and Brownian motion, Prog. Theor. Phys.33, 423 (1965)

    1965

  8. [16]

    Zwanzig,Nonequilibrium Statistical Mechanics(Oxford University Press, 2001)

    R. Zwanzig,Nonequilibrium Statistical Mechanics(Oxford University Press, 2001)

    2001

  9. [17]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, 2002)

    2002

  10. [18]

    Fundamental aspects of quantum Brownian motion,

    P. H¨ anggi and G.-L. Ingold, “Fundamental aspects of quantum Brownian motion,”Chaos15, 026105 (2005)

    2005

  11. [19]

    Multi-scale projection operator method and coarse- graining of covariant Fokker-Planck theory

    Ding, Mingnan, and Xiangjun Xing. “Multi-scale projection operator method and coarse- graining of covariant Fokker-Planck theory.” Physical Review Research 5.1 (2023): 013193

    2023

  12. [20]

    P. G. Bergmann and J. L. Lebowitz, New approach to nonequilibrium processes, Phys. Rev. 99, 578 (1955)

    1955

  13. [21]

    Stochastic processes: Time evolution, symmetries and linear response

    H¨ anggi, Peter, and Harry Thomas. “Stochastic processes: Time evolution, symmetries and linear response.” Physics Reports 88.4 (1982): 207-319

    1982

  14. [22]

    First and second law of thermodynamics at strong coupling

    Seifert, Udo. “First and second law of thermodynamics at strong coupling.” Physical review letters 116.2 (2016): 020601

    2016

  15. [23]

    ”Open system trajectories specify fluctuating work but not heat.” Physical Review E 94.2 (2016): 022143

    Talkner, Peter, and Peter H¨ anggi. ”Open system trajectories specify fluctuating work but not heat.” Physical Review E 94.2 (2016): 022143

    2016

  16. [24]

    Stochastic and macroscopic thermodynamics of strongly coupled systems

    Jarzynski, Christopher. “Stochastic and macroscopic thermodynamics of strongly coupled systems.” Physical Review X 7.1 (2017): 011008

    2017

  17. [25]

    Stochastic thermodynamics in the strong coupling regime: An unambiguous approach based on coarse graining

    Strasberg, Philipp, and Massimiliano Esposito. “Stochastic thermodynamics in the strong coupling regime: An unambiguous approach based on coarse graining.” Physical Review E 95.6 (2017): 062101. 30

    2017

  18. [26]

    Comment on ‘Measurability of nonequilibrium thermodynamics in terms of the Hamiltonian of mean force’

    P. Talkner and P. Hanggi. “Comment on ‘Measurability of nonequilibrium thermodynamics in terms of the Hamiltonian of mean force’ ” Phys. Rev E 21, 066101 (2020)

    2020

  19. [27]

    Colloquium:Statistical Mechanics and Thermodynamics at Strong Coupling: Quantum and Classical

    Talkner, Peter, and Peter Hanggi. “Colloquium:Statistical Mechanics and Thermodynamics at Strong Coupling: Quantum and Classical.” Rev. Mod. Phys. 92, 041002 (2020)

    2020

  20. [28]

    Thermodynamics and stochastic thermodynamics of strongly coupled systems

    Xing, Xiangjun, and Mingnan Ding. “Thermodynamics and stochastic thermodynamics of strongly coupled systems.” Physical Review E 109.3 (2024): 034105

    2024

  21. [29]

    Strong coupling thermodynamics and stochastic thermodynamics from the unifying perspective of time-scale separation

    Ding, Mingnan, Zhanchun Tu, and Xiangjun Xing. “Strong coupling thermodynamics and stochastic thermodynamics from the unifying perspective of time-scale separation.” Physical Review Research 4.1 (2022): 013015

    2022

  22. [31]

    Time-reversal and entropy

    Maes, Christian, and Karel Netocny. “Time-reversal and entropy.” Journal of statistical physics 110 (2003): 269-310

    2003

  23. [32]

    Local Detailed Balance.arXiv:2011.09200

    Maes, C. Local Detailed Balance.arXiv:2011.09200

    2011 arXiv

  24. [33]

    Quantum liquids: Bose condensation and Cooper pairing in condensed- matter systems

    Leggett, Anthony J. Quantum liquids: Bose condensation and Cooper pairing in condensed- matter systems. Oxford university press, 2006

    2006

  25. [34]

    Unified theory of thermodynamics and stochas- tic thermodynamics for nonlinear Langevin systems driven by non-conservative forces

    Ding, Mingnan, Fei Liu, and Xiangjun Xing. “Unified theory of thermodynamics and stochas- tic thermodynamics for nonlinear Langevin systems driven by non-conservative forces.” Phys- ical Review Research 4.4 (2022): 043125

    2022

  26. [35]

    Covariant nonequilibrium thermodynamics from Ito- Langevin dynamics

    Ding, Mingnan, and Xiangjun Xing. “Covariant nonequilibrium thermodynamics from Ito- Langevin dynamics.” Physical Review Research 4.3 (2022): 033247

    2022

  27. [36]

    Time-slicing path-integral in curved space

    Ding, Mingnan, and Xiangjun Xing. “Time-slicing path-integral in curved space.” Quantum 6 (2022): 694

    2022

  28. [37]

    Talkner, P

    P. Talkner, P. H¨ anggi, and M. Morillo,Microcanonical quantum fluctuation theorems, Phys. Rev. E77, 051131 (2008)

    2008

  29. [38]

    Talkner, M

    P. Talkner, M. Campisi, and P. H¨ anggi,Statistics of work and fluctuation theorems for mi- crocanonical initial states, New J. Phys.15, 095001 (2013)

    2013

  30. [39]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore,Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys.83, 863 (2011)

    2011

  31. [40]

    Goold, M

    J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk,The role of quantum information in thermodynamics—a topical review, J. Phys. A: Math. Theor.49, 143001 (2016). 31

    2016

  32. [41]

    Vinjanampathy and J

    S. Vinjanampathy and J. Anders,Quantum thermodynamics, Contemp. Phys.57, 545 (2016)

    2016

  33. [42]

    Gogolin and J

    C. Gogolin and J. Eisert,Equilibration, thermalisation, and the emergence of statistical me- chanics in closed quantum systems, Rep. Prog. Phys.79, 056001 (2016)

    2016

  34. [43]

    ”Thermodynamic laws in isolated systems.” Physical Review E 90.6 (2014): 062116

    Hilbert, Stefan, Peter Hanggi, and Jorn Dunkel. ”Thermodynamic laws in isolated systems.” Physical Review E 90.6 (2014): 062116

    2014

  35. [44]

    Hanggi, Peter, Stefan Hilbert, and Jorn Dunkel. ”Meaning of temperature in different ther- mostatistical ensembles.” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374.2064 (2016): 20150039

    2016

  36. [45]

    Frenkel, Daan, and Patrick B. Warren. ”Gibbs, Boltzmann, and negative temperatures.” American Journal of Physics 83.2 (2015): 163-170

    2015

  37. [46]

    unconditional

    Some author prefer an “unconditional” version of the LDB condition, in which both the forward and backward processes are assigned fixed initial distributions. While this approach is convenient for deriving certain fluctuation theorems, it introduces additional complications re...

  38. [47]

    Then ast→ ∞,U(t,0)A(z) stays in the same energy shell, and eventually spread uniformly, in the coarse-grained sense, to the entire energy shell

    In the special case of fixed control parameter and vanishing non-conservative force, the system dynamics is conserved and stationary. Then ast→ ∞,U(t,0)A(z) stays in the same energy shell, and eventually spread uniformly, in the coarse-grained sense, to the entire energy shell...

  39. [48]

    4.3 of Ref

    Note that our definition of GDB is different from that in Sec. 4.3 of Ref. [19]. 32

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.