Nonequilibrium Thermodynamics

(or irreversible thermodynamics), a general theory of the macroscopic description of nonequilibrium, or irreversible, processes.

Classical thermodynamics studies reversible thermodynamic processes. For nonequilibrium processes it establishes only inequalities that indicate the possible direction of the processes. The main task of nonequilibrium thermodynamics is the quantitative study of nonequilibrium processes, in particular, the determination of their rates in relation to external conditions. Non-equilibrium thermodynamics regards systems in which nonequilibrium processes occur as continuous media and considers the state parameters of such systems to be field variables, that is, continuous functions of the spatial coordinates and time.

In order to carry out the macroscopic description of a nonequilibrium process, the system is represented as consisting of volume elements that are big enough to contain a very large number of molecules. The thermodynamic state of each individual volume element is characterized by temperature, pressure, and other parameters that are used in equilibrium thermodynamics but that here are dependent on the coordinates and time. The quantitative description of nonequilibrium processes in this approach consists in constructing balance equations for the volume elements on the basis of the laws of conservation of mass, momentum, and energy and in constructing the entropy balance equation and phenomenological equations of the processes in question.

The first and second laws of thermodynamics can be formulated for nonequilibrium processes by using the methods of non-equilibrium thermodynamics. In addition, these methods make it possible to obtain from general principles, without considering the details of the mechanism of molecular interactions, a complete system of transport equations—that is, equations of hydrodynamics, heat conduction, and diffusion—for simple and complex systems (for example, systems with chemical reactions between the components and systems in which electromagnetic forces are taken into account).

Law of conservation of mass. For a multicomponent system the rate of change of the mass of the Ath component in a volume element is equal to the mass flux ρ_kV_k into the volume, where ρ_k is the density and v_k the velocity of the component. The change in density for an infinitesimal element of volume is the negative divergence of the mass flux. Consequently, the balance equation for the mass of the kth component has the form ∂ρ_k/∂t = – div ρ_kV_k. For the overall density

the law of conservation has the analogous form ∂ρ/∂t = – div ρv, where v is the medium’s velocity, which is dependent on the coordinates and time. For the concentration of some component c_k = ρ_k/ρ, the law of conservation of mass

permits determination of the diffusion flux J_k = ρ_k (V_k – V);here,

is the total derivative with respect to time.

Law of conservation of momentum. A change in the momentum of a volume element can result from forces that are due to the gradient of internal stresses in the medium P_αβ and from external forces F_k. By applying the law of conservation of momentum to volume elements, the fundamental Navier-Stokes equations of hydrodynamics can be obtained:

Here, the V_α are the Cartesian components of the velocity v, and P_βα is the stress tensor.

Law of conservation of energy. The law of conservation of energy for volume elements is the first law of thermodynamics in nonequilibrium thermodynamics. Here it is taken into consideration that the total specific energy consists of the specific kinetic and specific potential energies in the force field F_k and the specific internal energy u, which is the energy of thermal motion of the molecules and the average energy of molecular interactions. A balance equation analogous to (1) is obtained for u. It follows from this equation that the rate of change of momentum density per particle ∂ρu/∂t is determined by the divergence of the flux densities of internal energy ρuv and heat J_q and by the work done by the internal stresses

and by the applied forces

Entropy balance equation. In nonequilibrium thermodynamics it is assumed that the local entropy s of a volume element is the same function of internal energy u, specific volume v = 1/ρ, and concentration c_k as in the state of total equilibrium. Consequently, all conventional thermodynamic equalities are valid for the local entropy. Together with the laws of conservation of mass, momentum, and energy, these principles make it possible to derive the entropy balance equation

Here, σ is the local entropy production per unit volume per unit time, and J_s is the entropy flux, which is expressed in terms of the heat flux, diffusion flux, and the part of the stress tensor that is associated with nonequilibrium processes (that is, in terms of the viscous stress tensor Π_αβ).

In contrast to mass, energy, and momentum, entropy is not conserved but increases with time in a volume element at the rate σ as a result of irreversible processes. Furthermore, entropy can change through flow into or out of a volume element, a circumstance that is not connected with irreversible processes. In non-equilibrium thermodynamics the principle of entropy increase is expressed by the rate of entropy production being positive (σ > 0).

The rate of entropy production σ is determined solely by irreversible processes, such as diffusion, thermal conduction, and viscosity. In fact,

where the J_i, are fluxes (for example, the diffusion flux J_k, heat flux J_q, or viscous stress tensor Π_αβ) and the X_i are the conjugate generalized forces, or affinities, that is, the gradients of the thermodynamic parameters that bring about deviation from the equilibrium state. To obtain a complete system of equations describing nonequilibrium processes, nonequilibrium thermodynamics expresses the fluxes of physical quantities in terms of the generalized forces by using phenomenological equations.

Phenomenological equations. Nonequilibrium thermodynamics proceeds on the assumption that where there are small deviations of a system from thermodynamic equilibrium the fluxes that arise are linearly dependent on the generalized forces and are described by phenomenological equations of the type

where the L_ik are called kinetic, phenomenological, or transport coefficients.

In “direct” processes the force X_k gives rise to the flux J_k. For example, a temperature gradient causes a heat flux (heat conduction), a concentration gradient produces a mass flux (diffusion), a velocity gradient causes a momentum flux (determines the viscosity), and an electric field gives rise to an electric current (electrical conduction). The kinetic coefficients characterizing these processes are proportional to the coefficients of thermal conductivity, diffusion, viscosity, and electrical conductivity, respectively; these coefficients are also referred to as kinetic, or transport, coefficients.

A force X_k may also give rise to a flux J_i, where i ≠ k. For example, a temperature gradient may bring about a mass flux in multicomponent systems; this phenomenon is known as thermal diffusion, or the Soret effect. In the Dufour effect, on the other hand, a concentration gradient causes a heat flux. Such processes represent coupling, or interference, effects and are characterized by the coefficient L_ik, i ≠ k.

By making use of the phenomenological equations, we can obtain the following expression for the rate of entropy production:

The Prigogine theorem states that, for fixed external conditions that impede the attainment of equilibrium, σ is at a minimum in stationary states. In the state of thermodynamic equilibrium, σ = 0. One of the basic theorems of nonequilibrium thermodynamics is the Onsager theorem, which asserts the symmetry of the kinetic coefficients in the absence of an external magnetic field and in the absence of rotation of the system as a whole: L_ik = L_ki.

Discontinuous systems. In the examples considered above, the thermodynamic parameters were continuous functions of the coordinates. Nonequilibrium systems are possible in which the thermodynamic parameters change discontinuously; such systems are said to be discontinuous, or heterogeneous. As an example, consider two gases enclosed in vessels connected by a capillary tube or membrane. If the temperatures T and chemical potentials μ of the gases in the vessels are not equal (T₁ > T₂ and μ₁ > μ₂), then the forces X_u and X_m, where

produce between the two vessels the mass and energy fluxes J_m and J_u where J_m = L₁₁X_m + L₁₂X_u and J_u = L₂₁X_m + L₂₂X_U. In this case nonequilibrium thermodynamics explains the occurrence of the thermomolecular pressure difference and the thermomechanical effect.

Since the fluxes and forces in the given example are scalar quantities, such processes may be said to be scalar. The fluxes and forces are vectors in the processes of diffusion, heat conduction, thermal diffusion, and the Dufour effect; accordingly, these processes may be referred to as vector processes. A viscous flow, with shear viscosity, may be called a tensor process, since the forces and fluxes are tensors. According to a theorem derived by P. Curie, in an isotropic medium the relations between forces and fluxes can be given by linear equations only if the forces and fluxes are of the same tensor dimensionality. In this case the phenomenological equations are greatly simplified.

Nonequilibrium thermodynamics provides a theoretical basis for the investigation of open systems and makes it possible to explain many nonequilibrium phenomena in conductors, such as thermoelectric effects, galvanomagnetic effects, and thermomagnetic effects. Nonequilibrium statistical thermodynamics, which bears the same relation to nonequilibrium thermodynamics as statistical thermodynamics does to thermodynamics, develops a statistical foundation for the laws of nonequilibrium thermodynamics and derives expressions for kinetic coefficients in terms of the parameters of the structure of matter.