# Onsager reciprocal relations

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Onsager reciprocal relations

In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.

"Reciprocal relations" occur between different pairs of forces and flows in a variety of physical systems. For example, consider fluid systems described in terms of temperature, matter density, and pressure. In this class of systems, it is known that temperature differences 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. What is remarkable is the observation that, when both pressure and temperature vary, temperature differences at constant pressure can cause matter flow (as in convection) and pressure differences at constant temperature can cause heat flow. Perhaps surprisingly, the heat flow per unit of pressure difference and the density (matter) flow per unit of temperature difference are equal. This equality was shown to be necessary by Lars Onsager using statistical mechanics as a consequence of the time reversibility of microscopic dynamics (microscopic reversibility). The theory developed by Onsager is much more general than this example and capable of treating more than two thermodynamic forces at once, with the limitation that "the principle of dynamical reversibility does not apply when (external) magnetic fields or Coriolis forces are present", in which case "the reciprocal relations break down".

Though the fluid system is perhaps described most intuitively, the high precision of electrical measurements makes experimental realisations of Onsager's reciprocity easier in systems involving electrical phenomena. In fact, Onsager's 1931 paper refers to thermoelectricity and transport phenomena in electrolytes as well-known from the 19th century, including "quasi-thermodynamic" theories by Thomson and Helmholtz respectively. Onsager's reciprocity in the thermoelectric effect manifests itself in the equality of the Peltier (heat flow caused by a voltage difference) and Seebeck (electrical current caused by a temperature difference) coefficients of a thermoelectric material. Similarly, the so-called "direct piezoelectric" (electrical current produced by mechanical stress) and "reverse piezoelectric" (deformation produced by a voltage difference) coefficients are equal. For many kinetic systems, like the Boltzmann equation or chemical kinetics, the Onsager relations are closely connected to the principle of detailed balance and follow from them in the linear approximation near equilibrium.

For his discovery of these reciprocal relations, Lars Onsager was awarded the 1968 Nobel Prize in Chemistry. The presentation speech referred to the three laws of thermodynamics and then added "It can be said that Onsager's reciprocal relations represent a further law making possible a thermodynamic study of irreversible processes." Some authors have even described Onsager's relations as the "Fourth law of thermodynamics".

## Example: Fluid system

### Thermodynamic potentials, forces and flows

The basic thermodynamic potential is internal energy. In a fluid system, changes in the energy density $\ u$ depend on changes in matter density ρ and entropy density $\ s$ in the following way: $\ du= T ds + \mu d\rho$

where $\ T$ is temperature and μ is a combination of pressure and chemical potential. This formula is simply an expression of the first law of thermodynamics, namely energy conservation, where the Tds term represents heat exchange, and the μdρ term represents (mechanical and chemical) work done. For non-fluid or more complex systems there will be a different collection of variables describing the work term, but the principle is the same. We can write $ds = \frac{1}{T} du - \frac{\mu}{T} d\rho.$

The extensive quantities $\ u$ and ρ are conserved and their flows satisfy continuity equations: $\partial_{t}u + \nabla \cdot \mathbf{J}_{u} = 0 \!$

and $\partial_{t}\rho + \nabla \cdot \mathbf{J}_{\rho} = 0 \!$,

where $\partial_{t}$ indicates the partial derivative with respect to time $\ t$ (that is, the local time-rate of change), and $\ \nabla \cdot \!$ indicates the divergence of the flux densities $\ \mathbf{J} \!$.

The above expression of the first law in terms of entropy change defines the conjugate variables of $\ u$ and ρ, which are $\frac{1}{T}$ and $-\frac{\mu}{T}$ and are intensive quantities analogous to potential energies; their gradients of are called thermodynamic forces as they cause flows of the corresponding extensive variables as expressed in the following equations. In the absence of matter flows, we have a version of Fourier's law $\mathbf{J}_{u} = k\, \nabla\frac{1}{T} \!$;

and, in the absence of heat flows, we have a version of Fick's law $\mathbf{J}_{\rho} = -k'\, \nabla\frac{\mu}{T} \!$,

where $\ \nabla$ now indicates the gradient.

### 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 $\ L$): $\mathbf{J}_{u} = L_{uu}\, \nabla\frac{1}{T} - L_{u\rho}\, \nabla\frac{\mu}{T} \!$

and $\mathbf{J}_{\rho} = L_{\rho u}\, \nabla\frac{1}{T} - L_{\rho\rho}\, \nabla\frac{\mu}{T}. \!$

The Onsager reciprocity relations state the equality of the cross-coefficients $\ L_{u\rho}$ and $\ L_{\rho u}$. Proportionality follows from simple dimensional analysis (i.e., both coefficients are measured in the same units of temperature times mass density).

## Abstract formulation

Let $x_1,x_2,\ldots,x_n$ denote fluctuations from equilibrium values in several thermodynamic quantities, and let $S(x_1,x_2,\ldots,x_n)$ be the entropy. Then, assuming the fluctuations are small (gaussian), the probability distribution function can be expressed as $w=Ae^{-\frac{1}{2}\beta_{ik}x_ix_k}$ (where we are using Einstein summation convention and βik is some symmetric matrix)

Using the quasi-stationary equilibrium approximation, that is, assuming that the system is only slightly non-equilibrium, we have $\dot{x}_i=-\lambda_{ik}x_k$

Suppose we define thermodynamic conjugate quantities as $X_i=-\frac{\partial S}{\partial x_i}$, which can also be expressed as linear functions (for small fluctuations): Xi = βikxk

Thus, we can write $\dot{x}_i=-\gamma_{ik}X_k$ where $\gamma_{ik}=\lambda_{ik}\beta^{-1}_{ik}$ are called kinetic coefficients

The principle of symmetry of kinetic coefficients or the Onsager's principle states that γ is a symmetric matrix, that is γik = γki

### Proof

Define mean values ξi(t) and Ξi(t) of fluctuating quantities xi and Xi respectively such that they take given values $x_1,x_2,\ldots$ at t = 0 Note that $\dot{\xi}_i(t)=-\gamma_{ik}\Xi_k$

Symmetry of fluctuations under time reversal implies that $\langle x_i(t)x_k(0)\rangle=\langle x_i(-t)x_k(0)\rangle = \langle x_i(0)x_k(t)\rangle$

or, with ξi(t), we have $\langle\xi_i(t)x_k\rangle=\langle x_i\xi_k(t)\rangle$

Differentiating with respect to t and substituting, we get $\gamma_{il}\langle\Xi_l(t)x_k\rangle=\gamma_{kl}\langle x_i\Xi_l(t)\rangle$

Putting t = 0 in the above equation, $\gamma_{il}\langle X_lx_k\rangle=\gamma_{kl}\langle X_lx_i\rangle$

It can be easily shown from the definition that $\langle X_ix_k\rangle=\delta_{ik}$, and hence, we have the required result.

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