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The '''MINimum Entropy Production principle''' (MINEP) is an approximate variational characterization of steady states for thermodynamically open systems maintained out of equilibrium. Originally formulated within the framework of linear irreversible thermodynamics ([[#Pr|Prigogine 1947]]), it was extended to stochastic kinetics, e.g., for close-to-equilibrium systems described by a Master equation ([[#KM|Klein and Meijer 1954]]). The MINEP is consistent yet different from other nonequilibrium variational principles like the '''MAXimum entropy production principle''' ([[#Pa|Paltridge 1979]]) or the '''Least dissipation principle''' ([[#OM|Onsager and Machlup 1953]]). Recent dynamical fluctuation theories provide a framework for their precise formulation, unification and systematic improvement.


== Entropy production in irreversible thermodynamics ==


Irreversible thermodynamics considers systems consisting of several subsystems, each of them being at thermal equilibrium but where ''global'' thermodynamic equilibrium is broken. Within that framework the '''total entropy''' is defined as the sum of equilibrium entropies over all subsystems, including the environment (usually represented in terms of boundary conditions) in case the system is still not isolated. There can even be a continuum of subsystems in which case we speak about ''local'' equilibrium, ''local'' temperature, pressure or chemical potential, and we introduce the entropy ''density''.

Due to gradients of thermodynamic parameters (like temperature or chemical potential) called '''thermodynamic forces''' $F_\alpha$ there exist flows between the subsystems (like those of energy or mass) called '''thermodynamic fluxes''' $J_\alpha$. Forces and fluxes are naturally paired with each other so that the '''entropy production''' measuring the increase rate of total entropy has the form
\[
  \sigma = \sum_\alpha J_\alpha F_\alpha\,.
\]
The '''second law of thermodynamics''' requires $\sigma \geq 0$.

''Linear'' irreversible thermodynamics assumes
\[
  J_\alpha = \sum_\gamma L_{\alpha\gamma} F_\gamma
\]
with '''linear response coefficients''' $L_{\alpha\gamma}$ satisfying the '''Onsager-Casimir reciprocity relations''' $L_{\alpha\gamma} = L^*_{\gamma\alpha}$. The star indicates time-reversal of the microscopic dynamics, e.g., it inverts the sign of velocities and magnetic fields. In '''"even"''' systems $L_{\alpha\gamma} = L^*_{\alpha\gamma}$ and then the matrix $[L_{\alpha\gamma}]$ is ''symmetric'' and ''positive-definite''.


=== Minimum entropy production ===

MINEP states that whenever an ''even'' system $(L_{\alpha\gamma} = L_{\gamma\alpha}$) is driven out of equilibrium by applying time-independent constraints on the thermodynamic forces, then it approaches the ''steady'' state characterized by the minimum of the entropy production functional $\sigma = \sum_{\alpha\gamma} L_{\alpha\gamma} F_\alpha F_\gamma$.

In particular, if some but not all of the thermodynamic forces $F_1,\ldots,F_m$ are ''externally constrained'' then the steady values of the remaining forces $F_{m+1},\ldots,F_n$ are such that the corresponding currents vanish,
\[
  J_\alpha = 0\,,\qquad \alpha = m+1,\ldots,n\,.
\]

A dynamical form of the MINEP states that if the steady state is initially perturbed then the system returns to stationarity along a trajectory on which the entropy production is ''decreasing'', i.e., that $\sigma(t)$ is a [[Lyapunov function]].
 

=== Example A: Heat conduction ===

For pure heat conduction,
\[
  \sigma = \int_\Omega J_Q \cdot \nabla \bigl( \frac{1}{T} \bigr)\,d V\,,\qquad
  J_Q = \kappa\, \nabla \bigl( \frac{1}{T} \bigr)
\]
with $\nabla (1/T)$ the thermodynamic force and $J_Q$ the heat flux.  They are related by the Fourier law where we assume that the thermal conductivity $\kappa > 0$ is ''temperature-independent.'' Fixing the temperature profile on the boundary $\partial\Omega$, the entropy production is minimal for the bulk temperature profile satisfying the boundary condition and the ''stationary'' heat equation $\nabla \cdot J_Q = 0$.

The ''time-dependent'' heat equation $C(\partial T / \partial t) + \nabla \cdot J_Q = 0$ (with $C > 0$ the heat capacity) describes relaxation to the steady state. Along its solutions, $d\sigma / d t < 0$, ''i.e.'', the entropy production is monotone decreasing in time.


=== Example B: Electrical networks ===

In a linear electrical network made only of resistors and batteries the entropy production is proportional to the Joule heat dissipated altogether in all resistors. The voltages over resistors are constrained by the second Kirchhoff law: their sum balances the sum of all electromotive forces along any closed loop. Under these constraints, the MINEP is equivalent to the first Kirchhoff law: the sum of incoming currents to every node is zero.

For example, two resistors with resistances $R_1$, $R_2$ in series with electromotive force $E$ yield the entropy production
\[
  \sigma = \frac{1}{T} \Bigl( \frac{U_1^2}{R_1} + \frac{U_2^2}{R_2} \Bigr)
\]
where the voltages are constrained by $U_1 + U_2 = E$. Thus, $\sigma$ takes its minimum for $U_1 / R_1 = U_2 / R_2$.

The network above obtains a smooth dynamics by adding capacitors parallel to the resistors. Then the steady state remains unchanged and the entropy production is monotone in time as it approaches that steady regime.


=== Breaking of MINEP ===

The MINEP is generally ''not'' valid for systems which are either ''non-linear'' or ''non-even''. 

* The steady voltages in Example B are ''temperature-independent'' and generally corresponding to the minimum of the total Joule heat. Hence, the MINEP manifestly breaks down whenever the resistors dissipate at ''different'' temperatures. The temperature gradient provides another thermodynamic force which here does ''not'' couple with the electrical current; however it enters the linear response coefficients $L_i = 1 / (R_i T_i)$, yielding ''non-linear'' effects beyond the scope of the linear theory ([[#Ja|Jaynes 1980]]). a similar situation occurs in Example A in the case where the thermal conductivity $\kappa(T)$ is ''temperature-dependent''.

* The principle can break down even at ''homogeneous'' temperature, e.g., for a resistance $R$ in series with conductance $L$ and driven by electromotive force $E$: the steady current is $I = E / R$  and hence it obviously does ''not'' minimize the entropy production $\sigma = R I^2 / T$ as a function of the current ([[#L|Landauer 1975]]). The point is that the current $I$ playing here the role of thermodynamic force is ''odd'' under time-reversal, violating the conditions.



== Entropy production in stochastic kinetics ==

Stochastic kinetics describes ''fluctuating'' and ''thermodynamically open'' systems which interact with an environment consisting of one or several ''equilibrium'' reservoirs. Physical states and [[Stochastic dynamical systems|dynamics]] of the system are described  by probability distributions on configurations, respectively their time sequences (trajectories). Steady states generally have ''non-Boltzmann'' distributions and ''non-zero'' currents, unless the attached reservoirs are all at ''mutual'' equilibrium. 
    

=== Local detailed balance ===

The hypothesis of ''local'' detailed balance identifies the inherent asymmetry of ''Markov'' transition rates with entropy changes. It is a stochastic counterpart of the ''time-reversal invariance'' property of microscopic [[Hamiltonian dynamics|Hamiltonian systems]].

'''Local detailed balance''' states that for every single transition $x \rightarrow y$ between two configurations of the system, the ''antisymmetric'' component of the logarithm of the '''transition rate''' $k(x,y)$ corresponds to the increase of entropy in the environment,
\[
  k_B \log\frac{k(x,y)}{k(y,x)} = \Delta S_\text{env}(x \rightarrow y)\,.
\]
The environment may consist of several reservoirs ''not'' at mutual equilibrium; only one of them may assist at each transition. 

The entropy change in the reservoir is determined from the heat and the particle flow. E.g., for a ''heat'' reservoir at temperature $T$ involved in the transition $x\rightarrow y$, $\Delta S_\text{env}(x \rightarrow y) = \Delta Q(x \rightarrow y) / T$ with $\Delta Q$ the heat ''leaving'' the system, equal to the change of energy in the reservoir.

* If the system is coupled to a ''single'' heat reservoir and driven by ''conservative'' forces with potential $E(x)$ then $\Delta S_\text{env}(x \rightarrow y) = [E(x) - E(y)] / T$, yielding the usual '''(global) detailed balance''' condition of ''equilibrium'' dynamics,
\[ 
  e^{-\beta E(x)} k(x,y) = e^{-\beta E(y)} k(y,x)\,,\qquad \beta = \frac{1}{k_B T}\,.
\] 

* The general ''global'' detailed balance condition requires that $\sum_{i=0}^n \Delta S_\text{env}(x_i \rightarrow x_{i+1}) = 0$ for any ''closed'' sequence of configurations $x_0 \rightarrow \ldots \rightarrow x_n \rightarrow x_{n+1} = x_0$.  Equivalently, $\Delta S_\text{env}(x \rightarrow y) = \psi(y) - \psi(x)$  for some function $\psi$ on configurations. ''Global'' detailed balance gets ''violated'' under ''non-equilibrium'' conditions, e.g., (i) if the system is driven by ''non-conservative'' forces, or (ii) when different transitions are assisted by different reservoirs which are ''not'' mutually at equilibrium.


=== Entropy production ===

The time-evolving state of a Markov system is described by a '''probability distribution''' $\rho$ on configurations (= occupation statistics within the ensemble interpretation) and its entropy is identified with the [[Entropy|Gibbs-Shannon entropy]]
\[
  S = -k_B \sum_x \rho(x) \log \rho(x)\,.
\]
The rate of entropy increase in the environment (or, from the point of view of the open system, the outgoing '''entropy flux''') derives from the totality of local fluxes $j(x,y) = \rho(x) k(x,y) - \rho(y) k(y,x)$  along all transitions, 
\[
  J_S = \frac{1}{2}
  \sum_{x,y} j(x,y)\, \Delta S_\text{env}(x \rightarrow y)\,,
\]
and the local fluxes close the '''Master equation'''
\[
  \frac{d\rho(x)}{d t} + \sum_y j(x,y) = 0\,.
\]

The '''entropy production''' is the source term in the system's entropy balance equation
\[
  \frac{d S}{d t} + J_S = \sigma\,.
\]
That is now a functional on the space of probability distributions,
\[
  \sigma(\rho) = \sum_{x,y} \rho(x) k(x,y) \log \frac{\rho(x) k(x,y)}{\rho(y) k(y,x)}\,,
\]
and it retains the bilinear structure in fluxes and forces, with thermodynamic forces of the form $F(x,y) = \log \frac{\rho(x) k(x,y)}{\rho(y) k(y,x)}$
along each transition channel.

* The functional $\sigma(\rho)$ is ''non-negative'', ''convex'', and its extension to the domain of all (unnormalized) distributions is ''homogeneous'',  $\sigma(\lambda \rho) = \lambda\,\sigma(\rho)$.

* If a Markov system with finitely many configurations is '''irreducible''' (i.e., if any two configurations can be connected via a sequence of allowed transitions) then $\sigma(\rho)$ is ''strictly convex'', with ''unique'' minimum at some $\rho_P > 0$ (called '''Prigogine distribution''' in the sequel). 


The '''stationary distribution''' $\bar\rho$ is the (normalized) solution to the ''steady-state'' Master equation $\sum_y \bar j(x,y) = 0$ with $\bar j(x,y) = \bar\rho(x) k(x,y) - \bar\rho(y) k(y,x)$ the '''stationary currents'''.

* In ''finite'' ''irreducible'' systems the stationary distribution ''exists'' and it is ''unique''.

* Moreover, if the condition of ''global'' detailed balance is satisfied then (i) $\bar\rho = \rho_P$, (ii) $\bar j(x,y) = 0$ and $\bar F(x,y) = 0$ for all transitions, and (iii) $\sigma(\bar\rho) = 0$. In this case the system is at ''thermodynamic equilibrium''.


=== Stochastic MINEP ===

The Markov stochastic model fits the general framework of irreversible thermodynamics with $F(x,y)$ and $j(x,y)$ the conjugated forces and fluxes, both dependent on the probability distribution $\rho$. The ''linear'' relations between the forces and fluxes are (approximately) verified whenever the ''global'' detailed balance condition is only ''slightly violated''. In this linear regime the MINEP holds true and the ''Prigogine'' distribution $\rho_P$ well approximates the ''stationary'' distribution $\bar\rho$ ([[#KM|Klein and Meijer 1954]]; [[#JBN|Jiu-li et al 1984]]). 

Alternatively, in Schnakenberg's graph-theoretic representation the conjugated thermodynamic forces and fluxes are associated with independent cycles in the graph of allowed transitions ([[#S|Schnakenberg 1976]]). Those forces, defined by summing $F(x,y)$ over all transitions $x \rightarrow y$ along each cycle, are $\rho-$independent and hence provide an explicit realization of ''external constraints'' for the stochastic MINEP.

For an analytic formulation let the transition rates $k^\lambda(x,y)$ be ''smoothly'' dependent on $\lambda$ around $\lambda = 0$ with $k^0(x,y)$ corresponding to an ''irreducible'' system satisfying the ''global'' detailed balance condition. Then, $\bar\rho^\lambda - \rho_P^\lambda = O(\lambda^2)$, i.e., the ''stationary'' distribution and the ''Prigogine'' distribution coincide up to ''linear'' order in $\lambda$.


=== Derivation from dynamical fluctuations ===

The '''empirical occupation measure''' $p_\tau^\omega$ is the probability-measure-valued functional on the system's trajectories $(\omega_t)_{t=0}^\infty$ such that $p_\tau^\omega(x)$ equals the fraction of time in the interval $[0,\tau]$ that the trajectory $\omega$ visits the configuration $x$.

''Finite irreducible'' Markov systems are '''ergodic''' which means that $p_\tau^\omega \to \bar\rho$ in the limit $\tau \to \infty$, for ''almost every'' trajectory $\omega$.

According to the '''large-deviation principle''' for such Markov dynamics, the fluctuations of the empirical occupation measure have the exponential large-time asymptotics ([[#DV|Donsker and Varadhan 1975]])
\[
  \text{Prob}(p_\tau^\omega = \rho) \asymp e^{-\tau\, \Phi(\rho)}\,,\qquad
  \tau \to \infty\,.
\]
* The '''rate function''' $\Phi(\rho) \geq 0$ is ''strictly convex'' and $\Phi(\rho) = 0$ if and only if $\rho = \bar\rho$, the ''stationary'' distribution.

The minimization of $\Phi(\rho)$ is an ''exact'' stationary variational principle which for systems only slightly violating the ''global'' detailed balance condition boils down to minimizing the entropy production ([[#MN|Maes and Netočný 2007]]). It is an instance of general statistical relations between ''time-symmetric'' quantities (like the empirical occupation measure) and ''time-antisymmetric'' ones (like currents or entropy changes) which follows from the time-reversal symmetry of the underlying ''equilibrium'' dynamics. 

Analytically, if the transition rates $k^\lambda(x,y)$ are a smooth deformation of a ''finite irreducible'' system with $k^0(x,y)$ satisfying the ''global'' detailed balance condition, then 
\[
  \Phi^\lambda(\rho) = \frac{\sigma^\lambda(\rho) - \sigma^\lambda(\bar\rho^\lambda)}{4} + O(\lambda^2 d[\rho,\bar\rho^\lambda], d[\rho,\bar\rho^\lambda]^3)
\]
where the superscript $\lambda$ indicates  correspondence to the transition rates $k^\lambda(x,y)$, and $d[\rho,\bar\rho^\lambda]$ is the distance between $\rho$ and ''stationary'' distribution $\bar\rho^\lambda$. Whenever the higher-order corrections are ''negligible'' then the ''stationary'' distribution $\bar\rho^\lambda$ (i.e., the minimizer of $\Phi^\lambda$) coincides with the ''Prigogine'' distribution $\rho_P^\lambda$ (i.e., the minimizer of $\sigma^\lambda$).



== Entropy production in diffusive systems ==

The dynamics of a Brownian particle driven by force $f$ and moving in a medium characterized by temperature $T$ and ''positive-definite'' diffusion and mobility matrices $D$ and $\chi$, is in the high-friction limit given by the '''Smoluchowski equation'''
\[
  \frac{\partial\rho}{\partial t} + \nabla \cdot J = 0\,,\qquad J = \chi f \rho - D\nabla\rho
\]
with $\rho$ the '''probability density''' and $J$ the '''current density'''. Their ''stationary'' values $\bar\rho$ and $\bar J$ solve the ''steady-state'' equation $\nabla \cdot \bar J = 0$. 

* ''Local'' detailed balance corresponds to the '''Einstein relation''' $D = \chi k_B T$ between the diffusion and mobility matrices. (Let it be always assumed in the sequel.)

* ''Global'' detailed balance is satisfied if (i) the Einstein relation is valid and (ii) the force $f$ is ''conservative'', i.e., $f = -\nabla U$ for some (''globally'' defined) '''potential''' $U$. Provided $Z = \int_\Omega e^{-\beta U}\,dV < \infty$, the steady state with $\bar\rho = e^{-\beta U}/Z$ and  $\bar J = 0$ corresponds to thermodynamic equilibrium ($1/\beta = k_B T$).

* ''Violation'' of the ''global'' detailed balance, e.g., for the particle driven by a constant force along a circle, manifests the presence of ''non-equilibrium'' constraints which do not allow the particle to reach thermodynamic equilibrium.

The heat flux balances the power of driving force, yielding the entropy flux (i.e., the rate of entropy increase in the medium) $J_S = \beta \int_\Omega f \cdot J\,dV$. The '''entropy production''' term in the balance equation for the Gibbs-Shannon entropy $S = -\int \rho \log \rho\,dV$ is
\[
  \sigma = \int_\Omega J \cdot (\rho D)^{-1} J\,dV\,.
\]

=== MINEP for diffusion ===

The MINEP can be applied whenever the ''non-conservative'' component $\varepsilon \tilde f$ of the force $f^\varepsilon = -\nabla U + \varepsilon \tilde f$ is ''small''.

More precisely: As a functional on densities, $\sigma^\varepsilon(\rho)$ attains its minimum at $\rho_P^\varepsilon = \bar\rho^\varepsilon + O(\varepsilon^2)$, where $\bar\rho^\varepsilon$ is the ''stationary'' density for the particle driven by force $f^\varepsilon$. 

* The statement can be either (i) checked by a direct calculation, or (ii) verified by casting the model into the framework of irreversible thermodynamics with $X = \beta f - \nabla\log\rho$ the thermodynamic force conjugated to $J$, or (iii) proven to emerge from the general structure of dynamical fluctuations ([[#MNW|Maes et al 2008]]).

* The assumption of high-friction limit leading to the Smoluchowski equation is crucial here: the MINEP is generally ''not'' valid for the diffusion ''in phase space'' modeled by the Kramers equation since that is a ''non-even'' system (the momentum is ''odd'' under time-reversal).

There is also a '''modified MINEP for diffusions''' which remains ''exact'' at arbitrary distance from ''global'' detailed balance: With the driving force of the form $f = -\nabla U + \tilde f$ and taking the entropy production $\sigma(U)$ as a functional on the space of ''potentials'' $U$ (keeping $\rho$, $\tilde f$, $T$, and $D$ fixed), the $\sigma(U)$ attains its minimum at $U = U^*$ for which $\rho$ is ''stationary'', i.e.,
\[
  \nabla \cdot J^* = 0\,,\qquad J^* = \beta D (f - U^*)\rho - D\nabla\rho\,.
\]
This is a variational principle for the (''inverse'') ''stationary'' problem with variable $U^*$ instead of $\rho$.

* That follows from the ''convexity'' of $\sigma(U)$ and from the functional relation $\frac{\delta\sigma(U)}{\delta U} = 2\beta\,\nabla \cdot J$.

The formalism and the statements above have extensions to other diffusive systems, e.g., to diffusive fields in fluctuating hydrodynamics where $\rho$ becomes a density profile, the diffusion matrix $D(\rho)$ is $\rho-$dependent, and for various types of boundary conditions.


== References ==

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* Donsker, M. D. and Varadhan, S. R. S. 1975. Asymptotic evaluation of certain Markov process expectations for large time, I. ''Commun. Pure Appl. Math.'' 28: 1-47.
</span>

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* de Groot, S. R. and Mazur, P. 1962. ''Non-Equilibrium Thermodynamics''. Amsterdam, NorthHolland.
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* Eyink, G., Lebowitz, J. L. and Spohn, H. 1990. Macroscopic origin of hydrodynamic behavior: Entropy production and the steady state. In Campbell, D. (ed.) ''Chaos, Soviet-American Perspectives in Nonlinear Science'', American Institute of Physics, New York,  pp. 367–391.
</span>

<span id="Ja">
* Jaynes, E. T. 1980. The minimum entropy production principle. ''Ann. Rev. Phys. Chem.'' 31: 579--601.
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<span id="JBN">
* Jiu-li, L., Van den Broeck, C. and Nicolis, Z. 1984. Stability Criteria and Fluctuations around Nonequilibrium States. ''Z. Phys. B Condensed Matter'' 56: 165-170.
</span>

<span id="KM">
* Klein, M. J. and Meijer, P. H. E. 1954. Principle of minimum entropy production. ''Phys. Rev.'' 96: 250-255.
</span>

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* Landauer, R. 1975. Inadequacy of entropy and entropy derivatives in characterizing the steady state. ''Phys. Rev. A'' 12: 636-638.
</span>

<span id="MN">
* Maes, C. and Netočný, K. 2007. Minimum entropy production principle from a dynamical ﬂuctuation law. ''J. Math. Phys.'' 48: 053306.
</span>

<span id="MNW">
* Maes, C., Netočný, K. and Wynants B. 2008. Steady state statistics of driven diffusions. ''Physica A'' 387: 2675–2689.
</span>

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* Onsager, L. and Machlup S. 1953. Fluctuations and Irreversible Processes. ''Phys. Rev.'' 91: 1505-1512.
</span>

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* Paltridge, G. W. 1979. Climate and thermodynamic systems of maximum dissipation. ''Nature'' 279: 630-631.
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* Prigogine, I. 1947. ''Etude Thermodynamique des phénoménes irréversibles''. Desoer, Liége.
</span>

<span id="S">
*Schnakenberg, J. 1976. Network theory of behavior of Master equation systems. ''Rev. Mod. Phys.'' 48: 571-585.
</span>


=== Internal references ===

* Tomasz Downarowicz (2007) [[Entropy]]. Scholarpedia, 2(11):3901.
* Andre Longtin (2010) [[Stochastic dynamical systems]]. Scholarpedia, 5(4):1619.
* James Meiss (2007) [[Hamiltonian systems]]. Scholarpedia, 2(8):1943.


== See also ==

[[Fluctuations]],
[[Fluctuation theorem]],
[[Time's arrow and Boltzmann's entropy]]


[[Category:Statistical Mechanics]]