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. Thermodynamics (from the Greek thermos meaning heat and dynamis meaning power) is a branch of physics that studies the effects of temperature on physical systems at the macroscopic scale. ... In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks. ... This article may be too technical for most readers to understand. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann distribution. ...

As an example, it is observed that temperature differences in a system lead to heat flows from the warmer to the colder parts of the system. Similarly, pressure differences will lead to matter flow from high-pressure to low-pressure regions. It was observed experimentally that when both pressure and temperature vary, pressure differences can cause heat flow and temperature differences can cause matter flow. Even more surprisingly, the heat flow per unit of pressure difference and the density (matter) flow per unit of temperature difference are equal. This was shown to be necessary by Lars Onsager using statistical mechanics. Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... A red-hot iron rod cooling after being worked by a blacksmith. ... Pressure (symbol: p) is the force per unit area acting on a surface in a direction perpendicular to that surface. ... Matter is commonly referred to as the substance of which physical objects are composed. ... In the scientific method, an experiment is a set of actions and observations, performed to support or falsify a hypothesis or research concerning phenomena. ... Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ... Lars Onsager (November 27, 1903 – October 5, 1976) was a Norwegian physical chemist, winner of the 1968 Nobel Prize in Chemistry. ... 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. ...

Similar "reciprocal relations" occur between different pairs of forces and flows in a variety of physical systems.

The theory developed by Onsager is much more general than this example and capable of treating more than two thermodynamic forces at once.

Example: Fluid system

Thermodynamic potentials, forces and flows

The basic thermodynamic potential is internal energy. In a fluid system, the energy density depends on matter density and entropy density in the following way: 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... A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ...

where is temperature and is a combination of pressure and chemical potential. We can write The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume held fixed. ...

.

The extensive quantities and are conserved and their flows satisfy continuity equations: 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. ... Note that all the examples given below express the same idea (i. ...

and

,

where indicates the partial derivative with respect to time , and indicates the divergence of the flux densities . In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. ... 8:17 am, August 6, 1945, Japanese time. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

The gradients of the conjugate variables of and , which are and are thermodynamic forces and they cause flows of the corresponding extensive variables. In the absence of matter flows,

;

and, in the absence of heat flows,

,

where now indicates the gradient. 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 reciprocity relations

In this example, when there are both heat and matter flows, there are "cross-terms" in the relationship between flows and forces (the proportionality coefficients are customarily denoted by ):

and

.

The Onsager reciprocity relations state the equality of the cross-coefficients and . Proportionality follows from simple dimensional analysis (i.e., both coefficients are measured in the same units of temperature times mass density). Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... Measurement is the determination of the size or magnitude of something. ...

Abstract formulation

Let be the extensive variables on which entropy depends. In the following analysis, these symbols will refer to densities of these thermodynamic quantities. Then,

where

defines the intensive quantity conjugate to the extensive quantity .

The gradients of the intensive quantities are thermodynamic forces:

and they cause fluxes of the extensive quantities satisfying continuity equations

The fluxes are proportional to the thermodynamic forces by a matrix of coefficients

Then,

Introducing a susceptibility matrix

we have