Non-equilibrium thermodynamics is a branch of thermodynamics concerned with studying time-dependent thermodynamic systems, irreversible transformations and open systems. Non-equilibrium thermodynamics, as contrasted with equilibrium thermodynamics, is most successful in the study of stationary states, where there are nonzero forces, flows and entropy production, but no time variation. This article needs to be cleaned up to conform to a higher standard of quality. ... 8:17 am, August 6, 1945, Japanese time. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... Equilibrium Thermodynamics (Latin: aequalis = level and libra = weight or balance + Greek: thermos = heat and dynamis = power) is the systematic study of transformations of matter and energy in systems as they approach equilibrium. ... The thermodynamic entropy S (simply entropy in the context of thermodynamics) of a system is a measure of the uniqueness of its state in terms of its energy. ...

Basic concepts

The basic thermodynamic potential in equilibrium thermodynamics is, depending on the conditions, the internal energy (U) or a variation such as enthalpy (H = U + PV), Helmholz free energy (F = U - TS) or Gibbs free energy (G = U + PV - TS). However, in non-equilibrium thermodynamics it is entropy (S) that takes center stage. Irreversible transformations are characterized by net entropy production. In thermodynamics, four quantities, measured in units of energy, are called thermodynamic potentials: where T = temperature, S = entropy, p = pressure, V = volume Differential definitions The following differential relations hold for the four potentials: If we write the above four equations generally as Then it is seen that yielding expressions for... The internal energy of a system (abbreviated E or U) is the total kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the total potential energy associated with the vibrational and electric energy of atoms within molecules or crystals. ... Enthalpy (symbolized H, also called heat content) is the sum of the internal energy of matter and the product of its volume multiplied by the pressure. ... In thermodynamics, free energy is a measure of the amount of work that can be extracted from a system. ... Hi Chemistry kids! :-) In thermodynamics the Gibbs free energy is a state function of any system defined as G = H − T·S where G is the Gibbs free energy, measured in joules H is the enthalpy, measured in joules T is the temperature, measured in kelvins S is the entropy... The thermodynamic entropy S (simply entropy in the context of thermodynamics) of a system is a measure of the uniqueness of its state in terms of its energy. ...

Non-equilibrium thermodynamics applies to situations where the system under study is not in thermodynamic equilibrium but can be broken into subsystems which are sufficiently small to be in equilibrium, while still being large enough that thermodynamics is applicable to them. This hypothesis is known as local equilibrium. In some cases, there will be a discrete collection of systems interacting with each other through a discrete collection of channels. Continuous systems are studied by measuring extensive quantities per unit volume (as densities) and assuming that intensive quantities have locally defined values; this means that all thermodynamic variables can be represented by fields. Differences or gradients of intensive parameters are called thermodynamic forces, and they cause flows of the extensive variables. In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann distribution. ... In physics and chemistry, an extensive quantity (also referred to as an extensive variable) is a physical quantity whose value is proportional to the size of the system it describes. ... For other meanings of density, see density (disambiguation) Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ... In physics and chemistry, an intensive quantity (also referred to as an intensive variable) is a physical quantity whose value does not depend on the amount of the substance for which it is measured. ... Field (physics) - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... The flux visualized. ...

When an open system is allowed to reach a stationary state, it organizes itself so as to minimize total entropy production. This principle, emphasized by Ilya Prigogine among others, allows one to formulate stationary-state nonequilibrium thermodynamics using variational principles. Another powerful tool is provided by the Onsager reciprocal relations, which assert a certain symmetry between the response of two different flows to each other's thermodynamic forces. Ilya Prigogine (January 25, 1917 – May 28, 2003) was a Belgian physicist and chemist noted for his work on dissipative structures, complex systems, and irreversibility. ... A variational principle is a principle in physics which is expressed in terms of the calculus of variations. ... In thermodynamics, the Onsager reciprocal relations express the equality of certain relations between flows and forces in thermodynamical systems out of equilibrium, but where a notion of local equilibrium exists. ...

Flows and forces

Suppose that entropy S is given as a function of a collection of extensive variables E_i. Each extensive variable has a conjugate intensive variable called a thermodynamic force:

so that

dS = Σ_i I_i dE_i.

Each of the extensive variables E_i is assumed to be conserved. This means that the following continuity equations hold:

where J_i is the flux density of E_i.

It is possible to add source terms to the right-hand side if necessary.

Entropy production, the second law, and the Onsager relations

The time-variation of the entropy is then equal to

Here, Σ_i I_iJ_i is a reversible entropy flow (resulting in entropy thansfer through the boundaries of the system) and Σ_i ∇I_i · J_i is the rate of entropy production in the bulk.

In this context, the second law of thermodynamics can be stated as requiring that the rate of entropy production be nonnegative, that is, The second law of thermodynamics is a law of thermodynamics that states that all work tends towards the production of greater entropy over time. ...

Σ_i ∇I_i · J_i ≥ 0.

Otherwise, it would be possible to set up a configuration of thermodynamic forces and flows resulting in a decrease of entropy in an isolated system. This condition restricts what flows are possible in the pressence of given thermodynamic forces, without applying external work.

In the regime where both the flows are small and the thermodynamic forces vary slowly, there will be a linear relation between them, parametrized by a matrix of coefficients conventionally denoted L: For the square matrix section, see square matrix. ...

J_i = Σ_j L_ij ∇I_j.

The second law of thermodynamics requires that the matrix L be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that the matrix L is symmetric. This fact is called the Onsager reciprocal relations. In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In thermodynamics, the Onsager reciprocal relations express the equality of certain relations between flows and forces in thermodynamical systems out of equilibrium, but where a notion of local equilibrium exists. ...

Stationary states and the principle of minimal entropy production

Still needed.