The Bochkov-Kuzovlev, Jarzynski, and Crooks work theorems hold as exact statements in deterministic thermostatted dynamics only under two conditions derived by Kim (2015): a consistency condition requiring the phase-space contraction of all nonconservative forces to be offset by the dissipated power scaled by inverse temperature (which underpins the one-way exponential equalities), and a mesoscopic time-reversal parity condition on the momentum-dependent thermostatting force (which underpins the symmetric bidirectional Crooks and Bochkov-Kuzovlev relations). Realistic thermostatted systems such as linear damping and Gaussian isokinetic SLLOD shear flow violate the parity condition, and transitions between partial-equilibrium (metastable) states or nonequilibrium steady states evade the consistency condition. This problem asks for a general theory of work fluctuation relations for genuinely irreversible transformations in deterministic thermostatted dynamics: exact extensions to partial-equilibrium and steady-state transitions with phase-space contraction treated exactly, a delineation of the domain of applicability of the symmetric bidirectional relations in the reduced mesoscopic description, and quantitative robustness bounds when the consistency condition is satisfied only approximately over finite-time protocols. The core remains open: canonical-preserving deterministic thermostats were settled by Schöll-Paschinger and Dellago (2006), Markovian stochastic generalizations to arbitrary initial states and trajectory classes exist, and the deterministic steady-state-transition Crooks theorem is known only in an approximate large-protocol-duration limit (Patra and Bhattacharya 2016), but no exact deterministic thermostatted theory or finite-time robustness result has been found.

Background

Nonequilibrium work theorems relate the statistics of the work performed on a small driven system to equilibrium free-energy differences. The Jarzynski equality (JE) states that ⟨e−βW⟩=e−βΔF\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}, where WW is the work done on the system over a protocol of duration τ\tau, β=(kBT)−1\beta=(k_BT)^{-1}, and ΔF\Delta F is the free-energy difference between the equilibrium states at the protocol end points; the Crooks work fluctuation theorem (CWFT) refines this to a relation between forward and reverse work distributions, PF(W)/PR(−W)=eβ(W−ΔF)P_F(W)/P_R(-W)=e^{\beta(W-\Delta F)}; the Bochkov-Kuzovlev (BK) relation is the corresponding identity for Hamiltonian work in the system-centric picture. Kim (2015) formulated these theorems for deterministic dynamics of a thermally open system, i.e. a finite classical system coupled to a heat reservoir and externally driven, without appealing to a stochastic model. Two descriptions are used: the nonautonomous Hamiltonian picture, where the driving is built into the Hamiltonian as a time-dependent parameter, and the system-centric picture, where the time-dependent control forces Giex(t)G_i^{\mathrm{ex}}(t) appear directly in the equations of motion and the energy is the bare Hamiltonian. In the system-centric picture the extended equations of motion are q˙i=pi/m\dot{q}_i=p_i/m and p˙i=−∂V/∂qi+Di+Giex\dot{p}_i=-\partial V/\partial q_i+D_i+G_i^{\mathrm{ex}}, where Di(q,p)D_i(q,p) is a momentum-dependent dissipative force that represents the statistical correlation with the reservoir and plays the role of a deterministic thermostat. Such dynamics is generally non-Hamiltonian: the phase-space flow has nonzero compressibility Λ=∑i∂Di/∂pi\Lambda=\sum_i\partial D_i/\partial p_i, and the ensemble density obeys a generalized Liouville equation with a compressibility term. Writing Θ≡−∑iDiq˙i\Theta\equiv-\sum_i D_i\dot{q}_i for the dissipated power delivered to the reservoir (the Rayleigh dissipation function), Kim derived a consistency condition, ∫0τdt (Λ+βΘ)=0\int_0^\tau dt\,(\Lambda+\beta\Theta)=0 (the contracted phase volume of all nonconservative forces is exactly offset by β\beta times the dissipated power over the work period), under which the one-way exponential equalities (BK and JE) remain rigorous theorems; the condition is equivalent to detailed balance of the generalized Liouville dynamics and reduces, for linear damping Di=−γpiD_i=-\gamma p_i, to a nonequilibrium equipartition requirement on the time-averaged kinetic energy that becomes an identity in the equilibrium limit τ→∞\tau\to\infty but is a strong requirement over finite protocol periods. The symmetric bidirectional relations (BK work fluctuation theorem and CWFT, with WF=−WRW_F=-W_R) additionally require mesoscopic time-reversal invariance: the dissipative force must be an even function of momentum, D(−p)=D(p)D(-p)=D(p), the external fields must be symmetric under time inversion, and the compressibility factor must change sign under time reversal. Kim showed that common thermostatting mechanisms fail these parity conditions: linear damping is odd in momentum, and the Gaussian isokinetic (constant peculiar kinetic energy) SLLOD equations — the standard deterministic molecular-dynamics scheme for planar Couette shear flow — have a damping coefficient γ(p;β)\gamma(p;\beta) with no definite parity under momentum inversion, so the two-way relations do not apply and even the one-way theorem requires an exact cancellation between the momentum dependence of the thermostat and the shear dissipation. Adjacent known results bound the question. Schöll-Paschinger and Dellago (2006) proved the JE and CWFT for deterministic dynamics that conserves the canonical distribution — Hamiltonian dynamics, Nosé-Hoover dynamics (a deterministic thermostat with an auxiliary friction variable), Nosé-Hoover chains, and Gaussian isokinetic dynamics — via a relation between absorbed heat and the phase-flow Jacobian; this covers standard thermostat algorithms in the extended-variable description but not reduced descriptions without auxiliary variables, and not genuinely nonequilibrium thermostatted flows such as SLLOD shear. The Evans-Searles fluctuation theorem framework gives dissipation-function asymmetries for time-reversible deterministic dynamics in general. On the stochastic side, partial-equilibrium (metastable) branch free energies are accessible through fluctuation relations (Junier et al. 2009; Alemany et al. 2012; Alemany, Ribezzi-Crivellari and Ritort 2015), the JE, CWFT and heat theorems have been generalized to arbitrary initial distributions for Markovian systems (Gong and Quan 2015; experimental test by Hoang et al. 2018), and the trajectory-class fluctuation theorem of Wimsatt, Boyd and Crutchfield (2025) yields metastable memory-state free energies. What has no counterpart is the exact deterministic thermostatted theory when neither canonical preservation nor mesoscopic parity holds: irreversible transitions between partial-equilibrium or steady states, the domain of applicability of the symmetric relations in the reduced description, and the robustness of the equalities under approximate satisfaction of the consistency condition over finite-time protocols. The parallel stochastic and quantum formulations, which Kim's paper placed outside its scope, are mature fields (Seifert 2012; Campisi, Hänggi and Talkner 2011) and are not part of the open problem.

Problem Statement

Within the deterministic mesoscopic framework of Kim (2015) for a thermally open, externally driven classical system — extended equations of motion q˙i=pi/m\dot{q}_i=p_i/m, p˙i=−∂V/∂qi+Di(q,p)+Giex(t)\dot{p}_i=-\partial V/\partial q_i+D_i(q,p)+G_i^{\mathrm{ex}}(t) with a momentum-dependent thermostatting force DD, phase-space compressibility Λ=∑i∂Di/∂pi\Lambda=\sum_i\partial D_i/\partial p_i, and dissipated power Θ\Theta — the one-way exponential work equalities (Bochkov-Kuzovlev and Jarzynski) are guaranteed by the consistency condition

∫0τdt (Λ+βΘ)=0,\int_0^\tau dt\,(\Lambda+\beta\Theta)=0,

and the symmetric bidirectional relations (Bochkov-Kuzovlev work fluctuation theorem and Crooks relation, with WF=−WRW_F=-W_R) additionally require mesoscopic time-reversal invariance: D(−p)=D(p)D(-p)=D(p), external fields symmetric under time inversion, and Λ\Lambda odd under time reversal. The problem is to extend this theory to genuinely irreversible nonequilibrium transformations in deterministic thermostatted dynamics, with one overarching objective: determine which exponential work equalities remain valid, in what form, and with what accuracy, when these validity conditions no longer hold. This comprises three linked facets that jointly support that objective: (a) formulate and prove (or disprove) exact work fluctuation relations for irreversible transitions between partial-equilibrium (metastable) ensembles and between nonequilibrium steady states, for dynamics that neither preserves a canonical distribution nor satisfies the parity condition, with the phase-space contraction accounted for exactly rather than in a large-protocol-duration limit; (b) delineate the domain of applicability of the symmetric bidirectional work relations in the reduced mesoscopic description — for which dissipative force profiles D(q,p)D(q,p) on the bare phase space the two-way relation survives and for which it fails, i.e. a characterization covering both directions rather than a restatement of the necessary parity condition already given, and whether conjugate (time-reversed-protocol) or extended-variable (auxiliary-thermostat-variable) formulations restore it exactly in the cases where the reduced parity fails (linear damping, Gaussian isokinetic SLLOD shear flow); and (c) establish quantitative robustness bounds on the accuracy of the equalities when the consistency condition is satisfied only approximately over the protocol duration — as with the finite-time nonequilibrium equipartition condition for linear damping — so that the validity of experimental and computational applications can be assessed. All three facets (a)-(c) are jointly necessary for a complete solution: the objective is resolved only when the validity, form, and accuracy of the equalities are settled together, so a result confined to a single facet is partial progress toward the objective rather than a resolution. Progress on any facet counts toward the common objective only insofar as it contributes to the validity domain of the deterministic work theorems; the stochastic and quantum parallel formulations are excluded, since they constitute separate mature frameworks.

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