Liquid-Gas Transitions in Steady Heat Conduction

Non-equilibrium coexistence between a fluid and a hotter or colder crystal of granular hard disks

Non-equilibrium phase coexistence is commonly observed in both biological and artificial systems, yet understanding it remains a significant challenge. Unlike equilibrium systems, where free energy provides a unifying framework, the absence of such a quantity in non-equilibrium settings complicates their theoretical understanding. Granular materials, driven out of equilibrium by energy dissipation during collisions, serve as an ideal platform to investigate these systems, offering insights into the parallels and distinctions between equilibrium and non-equilibrium phase behavior. For example, the coexisting dense phase is typically colder than the dilute phase, a result usually attributed to greater dissipation in denser regions. In this article, we demonstrate that this is not always the case. Using a simple numerical granular model, we show that a hot solid and a cold liquid can coexist in granular systems. This counterintuitive phenomenon arises because the collision frequency can be lower in the solid phase than in the liquid phase, consistent with equilibrium results for hard-disk systems. We further demonstrate that kinetic theory can be extended to accurately predict phase temperatures even at very high packing fractions, including within the solid phase. Our results highlight the importance of collisional dynamics and energy exchange in determining phase behavior in granular materials, offering new insights into non-equilibrium phase coexistence and the complex physics underlying granular systems.

Direction of Spontaneous Processes in Non-Equilibrium Systems with Movable/Permeable Internal Walls

We consider three different systems in a heat flow: an ideal gas, a van der Waals gas, and a binary mixture of ideal gases. We divide each system internally into two subsystems by a movable wall. We show that the direction of the motion of the wall, after release, under constant boundary conditions, is determined by the same inequality as in equilibrium thermodynamics dU-đQ≤0. The only difference between the equilibrium and non-equilibrium laws is the dependence of the net heat change, đQ, on the state parameters of the system. We show that the same inequality is valid when introducing the gravitational field in the case of both the ideal gas and the van der Waals gas in the heat flow. It remains true when we consider a thick wall permeable to gas particles and derive Archimedes' principle in the heat flow. Finally, we consider the Couette (shear) flow of the ideal gas. In this system, the direction of the motion of the internal wall follows from the inequality dE-đQ-đWs≤0, where dE is the infinitesimal change in total energy (internal plus kinetic) and đWs is the infinitesimal work exchanged with the environment due to the shear force imposed on the flowing gas. Ultimately, we synthesize all these cases within a general framework of the second law of non-equilibrium thermodynamics.

Fundamental Relation for Gas of Interacting Particles in a Heat Flow

There is a long-standing question of whether it is possible to extend the formalism of equilibrium thermodynamics to the case of nonequilibrium systems in steady-states. We have made such an extension for an ideal gas in a heat flow. Here, we investigated whether such a description exists for the system with interactions: the van der Waals gas in a heat flow. We introduced a steady-state fundamental relation and the parameters of state, each associated with a single way of changing energy. The first law of nonequilibrium thermodynamics follows from these parameters. The internal energy U for the nonequilibrium states has the same form as in equilibrium thermodynamics. For the van der Waals gas, U(S*,V,N,a*,b*) is a function of only five parameters of state (irrespective of the number of parameters characterizing the boundary conditions): the effective entropy S*, volume V, number of particles N, and rescaled van der Waals parameters a*, b*. The state parameters, a*, b*, together with S*, determine the net heat exchange with the environment. The net heat differential does not have an integrating factor. As in equilibrium thermodynamics, the steady-state fundamental equation also leads to the thermodynamic Maxwell relations for measurable steady-state properties.

Nonintegral form of the reciprocal relation associated with violation of the fluctuation response relation

We extend Onsager's reciprocal relation to systems in a nonequilibrium steady state. While Onsager's reciprocal relation concerns the kinetic (Onsager) coefficient, the extended reciprocal relation concerns violation of the fluctuation response relation (FRR) for mechanical and thermal perturbations. This extended relation holds at each frequency when the extent of the FRR violation is expressed in a frequency domain. This nonintegral form distinguishes the extended relation from previous relations expressed by integration over a frequency. To obtain this relation, we consider one-particle one-dimensional systems described by an overdamped Langevin equation with a force driving the system away from equilibrium. We assume a special property of the potential in the system. From this Langevin equation, we obtain the Fokker-Planck (FP) equation describing the time evolution of the distribution function of the particle. Using the FP equation, we calculate the responses of the particle velocity and heat current by applying time-dependent perturbations of the driving force and temperature. We express the extent of the FRR violation in terms of these responses with time correlation functions and expand them in powers of the FP operator. This reciprocal relation is valid far from equilibrium. One can also confirm this reciprocal relation through experiments with systems such as colloidal suspensions because the FRR violation can be experimentally observed.

Control of Metastable States by Heat Flux in the Hamiltonian Potts Model

The local equilibrium thermodynamics is a basic assumption of macroscopic descriptions of the out of equilibrium dynamics for Hamiltonian systems. We numerically analyze the Hamiltonian Potts model in two dimensions to study the violation of the assumption for phase coexistence in heat conduction. We observe that the temperature of the interface between ordered and disordered states deviates from the equilibrium transition temperature, indicating that metastable states at equilibrium are stabilized by the influence of a heat flux. We also find that the deviation is described by the formula proposed in an extended framework of the thermodynamics.

Eigenvalue Crossing as a Phase Transition in Relaxation Dynamics

When a system's parameter is abruptly changed, a relaxation toward the new equilibrium of the system follows. We show that a crossing between the second and third eigenvalues of the relaxation operator results in a singularity in the dynamics analogous to a first-order equilibrium phase transition. While dynamical phase transitions are intrinsically hard to detect in nature, here we show how this kind of transition can be observed in an experimentally feasible four-state colloidal system. Finally, analytical proof of survival in the thermodynamic limit of a many body (1D Ising) model is provided.

Stochastic order parameter dynamics for phase coexistence in heat conduction

We propose a stochastic order parameter model for describing phase coexistence in steady heat conduction near equilibrium. By analyzing the stochastic dynamics with a nonequilibrium adiabatic boundary condition, where total energy is conserved over time, we derive a variational principle that determines thermodynamic properties in nonequilibrium steady states. The resulting variational principle indicates that the temperature of the interface between the ordered region and the disordered region becomes greater (less) than the equilibrium transition temperature in the linear response regime when the thermal conductivity in the ordered region is less (greater) than that in the disordered region. This means that a superheated ordered (supercooled disordered) state appears near the interface, which was predicted by an extended framework of thermodynamics proposed in Nakagawa and Sasa [Liquid-Gas Transitions in Steady Heat Conduction, Phys. Rev. Lett. 119, 260602 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.260602.].

Phase separation driven by production of architectural RNA transcripts

We here use an extension of the Flory-Huggins theory to predict that phase separation is driven by the production of architectural RNA (arcRNA) at a DNA locus with a constant rate. The arcRNA molecules diffuse in the nucleoplasm and show attractive interactions via proteins that are bound to the arcRNA. Our theory predicts that when the Flory interaction parameter is larger than the value at the critical point, the volume fraction of arcRNA jumps between the two values corresponding to the volume fraction of the two coexisting phases at equilibrium at a distance from the DNA locus due to the local equilibrium condition. The distance defines the radius of the condensate that is assembled by the phase separation. When the interaction parameter is large, the volume of the condensates is proportional to the production rate of arcRNA and inversely proportional to the degradation rate of arcRNA. These results imply that most arcRNA molecules are degraded before they diffuse out from the condensates due to the strong segregation of arcRNA.