Deterministic Thermal Reservoirs

Storage of Energy in Constrained Non-Equilibrium Systems

We study a quantity T defined as the energy U, stored in non-equilibrium steady states (NESS) over its value in equilibrium U0, ΔU=U−U0 divided by the heat flow JU going out of the system. A recent study suggests that T is minimized in steady states (Phys.Rev.E.99, 042118 (2019)). We evaluate this hypothesis using an ideal gas system with three methods of energy delivery: from a uniformly distributed energy source, from an external heat flow through the surface, and from an external matter flow. By introducing internal constraints into the system, we determine T with and without constraints and find that T is the smallest for unconstrained NESS. We find that the form of the internal energy in the studied NESS follows U=U0∗f(JU). In this context, we discuss natural variables for NESS, define the embedded energy (an analog of Helmholtz free energy for NESS), and provide its interpretation.

Backward and covariant Lyapunov vectors and exponents for hard-disk systems with a steady heat current

The covariant Lyapunov analysis is generalized to systems attached to deterministic thermal reservoirs that create a heat current across the system and perturb it away from equilibrium. The change in the Lyapunov exponents as a function of heat current is described and explained. Both the nonequilibrium backward and covariant hydrodynamic Lyapunov modes are analyzed and compared. The movement of the converged angle between the hydrodynamic stable and unstable conjugate manifolds with the free flight time of the dynamics is accurately predicted for any nonequilibrium system simply as a function of their exponent. The nonequilibrium positive and negative LP mode frequencies are found to be asymmetrical, causing the negative mode to oscillate between the two functional forms of each mode in the positive conjugate mode pair. This in turn leads to the angular distributions between the conjugate modes to oscillate symmetrically about π/2 at a rate given by the difference between the positive and negative mode frequencies.

Dissipation and entropy production in deterministic heat conduction of quasi-one-dimensional systems

We explore the consequences of a deterministic microscopic thermostat-reservoir contact mechanism. With different temperature reservoirs at each end of a two-dimensional system, a heat current is produced and the system has an anomalous thermal conductivity. The microscopic form for the local heat flux vector is derived and both the kinetic and potential contributions are calculated. The total heat flux vector is shown to satisfy the continuity equation. The properties of this nonequilibrium steady state are studied as functions of system size and temperature gradient, identifying key scaling relations for the local fluid properties and separating bulk and boundary effects. The local entropy density calculated from the local equilibrium distribution is shown to be a very good approximation to the entropy density calculated directly from the velocity distribution even for systems that are far from equilibrium. The dissipation and kinetic entropy production and flux are compared quantitatively and the differing mechanisms discussed within the Bhatnagar-Gross-Krook approximation. For equal-temperature reservoirs the entropy production near the reservoir walls is shown to be proportional to the local phase space contraction calculated from the tangent space dynamics. However, for unequal temperatures, the connection between local entropy production and local phase space contraction is more complicated.