Local relation between phase-space contraction and entropy production in deterministic-reservoir quasi-one-dimensional heat conduction

Deterministic thermal reservoirs turn thermostatting into a mechanical, time-reversible dynamical process: a quasi-one-dimensional (QOD) channel of hard disks whose end particles collide with reservoir walls under a deterministic momentum-reset rule. For this system Morriss and Truant (Phys. Rev. E 87, 062144 (2013)) showed that the local kinetic entropy production at the boundary particles is empirically proportional to the tangent-space phase-space contraction at the corresponding reservoir when the two reservoirs have equal temperatures — a particle-resolved link between dynamical-system dissipation and thermodynamic entropy production — but that this proportionality disappears for unequal temperatures, while all their entropy quantities omitted the configurational (positional-correlation) contributions that dominate at high density, and the entropy production itself was estimated with a BGK relaxation closure whose prefactor could not be consistently fixed. The open problem is to establish, or to refute with controlled evidence, the general local relation between reservoir dissipation and total entropy production in this setting: the explicit correction terms that appear for unequal temperatures, the microscopic inclusion of configurational entropy density, flux, and production, and a justified kinetic closure for the entropy production. Later work has advanced only fragments (a two-body configurational entropy density; a crossover in the conductivity exponent); the core relation remains open.

Background

Heat transport in many-particle systems couples two descriptions that are usually developed separately: thermodynamics, where irreversibility is quantified by an entropy production, and dynamical systems theory, where dissipation is quantified by the contraction of phase-space volumes. The problem asks for the microscopic bridge between the two, at the level of individual particles, in a model fluid simple enough that both sides can be computed essentially without approximation.

The model is a quasi-one-dimensional (QOD) hard-disk fluid: $N$ hard disks of diameter $\sigma$ confined to a channel of width $L_y$ between one and two disk diameters (in the source study $L_y=1.15\sigma$), so no disk can pass another and the left-to-right particle order is fixed forever. Because the ordering never changes, each particle occupies on average its own volume element, and local thermodynamic fields (density, kinetic temperature, entropy density) can be assigned to a single particle rather than to a coarse-grained cell. Such channels conduct heat anomalously: the effective thermal conductivity diverges with system size, as is typical of one-dimensional and quasi-one-dimensional systems, for example the one-dimensional hard-particle gas and narrow-channel hard-disk fluids studied earlier with stochastic reservoirs.

Most simulations of such systems use stochastic reservoirs that reinject particles with velocities drawn from a temperature-dependent distribution, so the reservoir contact is not itself a deterministic dynamical process. The source study instead uses deterministic thermal reservoirs, introduced for this quasi-one-dimensional geometry by Taniguchi and Morriss in 2007: when a disk collides with a reservoir wall, its tangential momentum component is unchanged and its normal component is reset according to

$$p_x' = \epsilon\, p_{\mathrm{res}} - (1-\epsilon)\, p_x,$$

where $p_{\mathrm{res}}=\sqrt{2T_{\mathrm{res}}}$ is a fixed reservoir momentum set by the reservoir temperature and $\epsilon\in[0,1]$ controls the coupling ($\epsilon\to 0$ gives a hard wall, $\epsilon\to 1$ fully replaces the incoming momentum). The whole dynamics, bulk and boundary, is then deterministic, so dynamical-systems diagnostics become available: the tangent-space (linearized) dynamics can be integrated through every collision, and the sum of the Lyapunov exponents equals the time-averaged sum over reservoir collisions of $\ln|d_I|$, where $d_I$ is the determinant of the linearized wall-collision matrix at reservoir $I$. Phase-space contraction arises only at reservoir collisions, and can be attributed to the left or right reservoir separately.

On the thermodynamic side, a local entropy balance $\partial s/\partial t + \partial j_s/\partial x = \sigma$ is built from the one-particle velocity distribution $f(x,\mathbf v)$: the kinetic entropy density and entropy flux follow from $-k\int f\ln f$, while the entropy production $\sigma$ requires a closure for the Boltzmann collision integral. The source uses the BGK (Bhatnagar–Gross–Krook) relaxation-time form

$$\sigma(x) = \nu(x)\, k\int d\mathbf v\,\{f(x,\mathbf v)-f_{\mathrm{loc}}(x,\mathbf v)\}\ln f(x,\mathbf v),$$

with $\nu(x)=\nu_0\, n(x)\sqrt{T(x)}$ and an unknown prefactor $\nu_0$. (The paper's abstract calls this the "BGY" approximation; the formula actually used is the standard BGK relaxation form, and the two labels denote the same closure in this paper.) The configurational part of the entropy — the part depending on positional correlations between particles rather than velocities — was omitted from all entropy quantities.

The source study established the following. A microscopic local heat-flux vector with kinetic and collisional (potential) parts satisfies the continuity equation and matches the energy fluxes through the reservoirs to within about $0.3\%$. The kinetic entropy density is closely approximated by an anisotropic Gaussian local-equilibrium form even far from equilibrium. A Clausius-like relation $j_s \approx J_Q/T$ between kinetic entropy flux and heat flux holds within roughly $10\%$ at interior particles at low density, but fails at high density, where the potential part carries more than $80\%$ of the heat flux at density $\rho=0.8$. For equal reservoir temperatures ($T_L=T_R$), the kinetic entropy production at the particles touching each reservoir is empirically proportional to the phase-space contraction at that reservoir when both are parametrized by the coupling strength $\epsilon$. For unequal temperatures, this proportionality disappears: the phase-space contractions are essentially linear in the reservoir temperatures, while the entropy productions are not, and the mismatch is worse at high density. The BGK prefactor fitted from the entropy balance, $\nu_0\approx 0.22$–$0.30$, does not match the value $3/2$ obtained by identifying the relaxation rate with the directly measured collision frequency, an identification that itself fails completely at high density.

A partial sequel by the same author computed the local configurational entropy density in the nonequilibrium channel from two-particle correlation functions: it is negative and smaller in magnitude than the positive kinetic part, so the total local entropy density remains positive. That computation, however, kept only the two-body term, explicitly left open whether the quantity constructed is the thermodynamically meaningful nonequilibrium entropy, and did not compute the configurational entropy flux or production, nor revisit the comparison with phase-space contraction.

Problem Statement

Establish — or refute with controlled evidence — the local, microscopic relation between dynamical-system dissipation and thermodynamic entropy production in the nonequilibrium steady state of the deterministic-reservoir quasi-one-dimensional hard-disk system, generalizing the equal-temperature kinetic proportionality observed by Morriss and Truant. A complete answer comprises three linked deliverables:

1. Unequal temperatures. Derive the relation between the tangent-space dissipation at each reservoir (the per-collision Jacobian determinants $d_I$ and their collision rates, whose averages give the contributions to the Lyapunov sum) and the local entropy production at the boundary particles, valid for $T_L \neq T_R$: either an equivalence with explicit, controlled correction terms — with an identified physical origin, such as the non-Gaussian velocity distributions in the boundary layers, the density and collision-frequency asymmetry between hot and cold reservoirs, or gradient-dependent corrections — or a demonstration that no local relation of this form exists, together with what replaces it (for example, only a global balance over both reservoirs). The source's equal-temperature observation — proportionality between the kinetic (BGK) entropy production of the boundary particles and the per-reservoir phase-space contraction at low density ($\rho\approx0.03$), parametrized by the coupling strength $\epsilon$ — must be recovered or explained in the $T_L \to T_R$ limit within its observed domain of validity; under the refutation branch, this empirical observation must be accounted for rather than reproduced by a general relation. The result must also be consistent with the source's measurements: phase-space contraction essentially linear in the reservoir temperature, entropy production not.

2. Configurational contributions. Close the entropy balance with the configurational (positional-correlation) part: microscopic expressions and particle-resolved evaluation of the configurational contributions to the local entropy density, the entropy flux, and the entropy production — extending the two-particle-correlation computation of the density to the flux and production, and controlling the neglected three-body and higher terms — so that, at high density where potential contributions dominate the heat flux, the dissipation comparison is made against the total entropy production rather than the kinetic part alone.

3. Closure control. Justify or replace the kinetic closure used for the entropy production: resolve why the entropy-balance fit gives $\nu_0\approx0.22$–$0.30$ while the collision-frequency identification gives $3/2$, determine the true spatial and gradient dependence of the relaxation rate in the nonequilibrium QOD channel, or replace the BGK form with a controlled collision operator valid for inhomogeneous QOD states — since the entropy production entering deliverables 1 and 2 is only defined through this closure.

These deliverables jointly support one objective: a quantitatively controlled identification of dynamical dissipation with thermodynamic entropy production at the contacts with mechanically coupled reservoirs. The problem is complete when that identification (or its refutation) is established across the density and temperature-gradient range of the model — spanning the kinetic-dominated ($\rho\approx0.03$) and potential-dominated ($\rho\approx0.8$) regimes — and when the source's equal-temperature kinetic proportionality is recovered or explained in the $T_L \to T_R$ limit within its observed low-density domain of validity. The two-regime span applies to the unequal-temperature relation, the configurational closure, and the closure control; no equal-temperature proportionality is required to hold at $\rho\approx0.8$ or against the total entropy production, since none was demonstrated there. An answer must not silently replace this objective with a narrower target, such as a purely global relation or a single-density study.

References

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