Generalized thermodynamic relations for a system experiencing heat and mass diffusion in the far-from-equilibrium realm based on steepest entropy ascent

This paper presents a nonequilibrium thermodynamic model for the relaxation of a local, isolated system in nonequilibrium using the principle of steepest entropy ascent (SEA), which can be expressed as a variational principle in thermodynamic state space. The model is able to arrive at the Onsager relations for such a system. Since no assumption of local equilibrium is made, the conjugate fluxes and forces are intrinsic to the subspaces of the system's state space and are defined using the concepts of hypoequilibrium state and nonequilibrium intensive properties, which describe the nonmutual equilibrium status between subspaces of the thermodynamic state space. The Onsager relations are shown to be a thermodynamic kinematic feature of the system independent of the specific details of the micromechanical dynamics. Two kinds of relaxation processes are studied with different constraints (i.e., conservation laws) corresponding to heat and mass diffusion. Linear behavior in the near-equilibrium region as well as nonlinear behavior in the far-from-equilibrium region are discussed. Thermodynamic relations in the equilibrium and near-equilibrium realm, including the Gibbs relation, the Clausius inequality, and the Onsager relations, are generalized to the far-from-equilibrium realm. The variational principle in the space spanned by the intrinsic conjugate fluxes and forces is expressed via the quadratic dissipation potential. As an application, the model is applied to the heat and mass diffusion of a system represented by a single-particle ensemble, which can also be applied to a simple system of many particles. Phenomenological transport coefficients are also derived in the near-equilibrium realm.

Figure 1

For mass diffusion, mass flow and energy flow are both allowed across the partition. For heat diffusion, only energy flow is allowed.

Figure 2

Thermodynamic trajectories on a temperature-particle number diagram. The initial probabilities for the two subsystems [black line for subsystem $a$, green (gray) line for subsystem $b]$ are the same, while the initial temperature of subsystem $b$ is 1500 K and that for subsystem $a$ is 300 K (solid line), 500 K (dashed line), 800 K (dashed-dotted line), and 1100 K (dotted line), respectively.

Figure 3

Thermodynamic trajectories on a temperature-particle number diagram for the three cases. Case 1: $p^a=0.8, p^b=0.2, T^a=1500$ K, and $T^b=300$ K (dashed-dotted line); case 2: $p^a=0.8, p^b=0.2, T^a=900$ K, and $T^b=900$ K (solid line); and case 3: $p^a=0.2, p^b=0.8, T^a=1500$ K, and $T^b=300$ K (dashed line). The black lines are the trajectory of subsystem $a$, and the green (gray) lines are the trajectory of subsystem $b$.

Figure 4

Entropy evolution and entropy generation rate in dimensionless time for the three cases of Fig. 3. Case 1: $p^a=0.8, p^b=0.2, T^a=1500$ K, and $T^b=300$ K (dashed-dotted line); case 2: $p^a=0.8, p^b=0.2, T^a=900$ K, and $T^b=900$ K (solid line); and case 3: $p^a=0.2, p^b=0.8, T^a=1500$ K, and $T^b=300$ K (dashed line). The black lines are the entropy evolutions relative to the vertical axis on the left, and the green (gray) lines are the entropy generation rates relative to the vertical axis on the right.

Figure 5

Entropy evolution and entropy generation rate in real time for the three cases of Fig. 3. Case 1: $p^a=0.8, p^b=0.2, T^a=1500$ K, and $T^b=300$ K (dashed-dotted line); case 2: $p^a=0.8, p^b=0.2, T^a=900$ K, and $T^b=900$ K (solid line); and case 3: $p^a=0.2, p^b=0.8, T^a=1500$ K, and $T^b=300$ K (dashed line). The black lines are the entropy evolutions relative to the vertical axis on the left, and the green (gray) lines are the entropy generation rates relative to the vertical axis on the right.