A Theoretical Basis for Maximum Entropy Production

Abstract

Maximum entropy production (MaxEP) is a conjectured selection criterion for the stationary states of non-equilibrium systems. In the absence of a firm theoretical basis, MaxEP has largely been applied in an ad hoc manner. Consequently its apparent successes remain something of a curiosity while the interpretation of its apparent failures is fraught with ambiguity. Here we show how Jaynes’ maximum entropy (MaxEnt) formulation of statistical mechanics provides a theoretical basis for MaxEP which answers two outstanding questions that have so far hampered its wider application: What do the apparent successes and failures of MaxEP actually mean physically? And what is the appropriate entropy production that is maximized in any given problem? As illustrative examples, we show how MaxEnt underpins previous applications of MaxEP to planetary climates and fluid turbulence. We also discuss the relationship of MaxEP to the fluctuation theorem and Ziegler’s maximum dissipation principle.

Appendices

Appendix A: Identifying the Lagrange Multiplier λ (Sect. 3.4.1)

In Sect. 3.4.1 (zonal energy balance models), the MaxEnt p.d.f. is p(f) ∝ e ^A where $A = \sum\nolimits_{i} {\left( {\upalpha \Updelta f_{i} + \uplambda_{i} f_{i} } \right)}$, where f _i is the meridional heat flux from zone i − 1 to zone i, ∆f _i  = f _i  − f _i+1 is the heat convergence into zone i, and $\upalpha$ and ${\uplambda_{i} }$ are the Lagrange multipliers associated with the global energy balance constraint $\sum\nolimits_{i} {\Updelta F_{i} = \, 0}$ and the trial flux F _i , respectively. Here we show that ${\uplambda_{i} }$ may be identified with the gradient in inverse temperature –∆(1/T _i ) = 1/T _i+1 – 1/T _i . Consider local heat balance over some finite time interval (0, τ). Let f _i and Δf _i denote time-averaged rates over ($0,\uptau$), and let u _i (t) denote the heat content of zone i at time t. Local heat balance then gives $\left\{ {u_{i} \left( \uptau \right) \, -u_{i} \left( 0 \right)} \right\}/\uptau = \Updelta f_{i} - r_{i}$ where r _i  = f _LW,i ↑ – F _SW,i ↓ is the net radiation leaving zone i, with ensemble average R _i  = ∆F _i . By introducing constraints on the ensemble averages of u _i (τ) in each zone, for which the corresponding Lagrange multipliers may be identified with⁸ −1/T _i we then have $A = \, -\sum\nolimits_{i} {u_{i} \left( \uptau \right)/T_{i} } + \sum\nolimits_{i} {\left( {\upalpha \Updelta f_{i} + \uplambda_{i} f_{i} } \right)}$ which from heat balance can be rewritten as $A = - \frac{1}{2}\sum\nolimits_{i} {\left\{ {u_{i} \left( 0 \right) + u_{i} \left( \uptau \right)} \right\}/T_{i} } + \sum\nolimits_{i} {\Updelta f_{i} \left( {\upalpha -\frac{1}{2}\uptau /T_{i} } \right)} + \frac{1}{2}\uptau \sum\nolimits_{i} {r_{i} /T_{i} } + \sum\nolimits_{i} {\uplambda_{i} f_{i} }$. Summing by parts reduces this to $A = - \frac{1}{2}\sum\nolimits_{i} {\left\{ {u_{i} \left( 0 \right) + u_{i} \left( \uptau \right)} \right\}} /T_{i} + \sum\nolimits_{i} {f_{i} \left( {\uplambda_{i} + \frac{1}{2} \uptau \Updelta \left( { 1/T_{i} } \right)} \right)} + \frac{1}{2} \uptau \sum\nolimits_{i} {r_{i} /T_{i}}.$ Since knowledge of the radiation fluxes R _i is equivalent to knowledge of the heat fluxes F _i , the second term in A is redundant in the presence of the third term, implying $\uplambda_{i} = -\frac{1}{2} \uptau \Updelta \left( { 1/T_{i} } \right)$ as we wished to show. Substituting this into Eq. (3.19) for the mean entropy production then gives $I = 2\sum\nolimits_{i} {\uplambda_{i} F_{i} \propto } -\sum\nolimits_{i} {F_{i} \Updelta \left( { 1/T_{i} } \right)} = \sum\nolimits_{i} {\Updelta F_{i} /T_{i} .}$

Appendix B: Identifying the Lagrange Multiplier α₁(z) (Sect. 3.4.2)

In Sect. 3.4.2.1 (shear turbulence in plane Couette flow), the MaxEnt p.d.f. is p(f) ∝ e ^A where $A = \left\langle {\upalpha_{ 1} \left( z \right)(\partial_{z} \overline{u}_{x} - \overline{{v_{x} v_{z} }} + \left\langle {v_{x} v_{z} } \right\rangle + Re)} \right\rangle + \upalpha_{ 2} (\left\langle {\partial_{j} u_{i} \partial_{j} u_{i} } \right\rangle - Re^{ 2} - Re\left\langle {v_{x} v_{z} } \right\rangle )$. Ignoring the fluctuation terms involving v _x v _z and setting $\varvec{u} = U(z)\hat{\varvec{x}}$ (mean-field approximation) yields $A = \int {{\text{d}}z\{ \upalpha_{ 1} \left( z \right)U^{\prime } \left( z \right) + \upalpha_{ 2} U^{\prime } \left( z \right)^{ 2} \} }$ where we have dropped the constant terms involving Re and Re ². The action A is stationary with respect to U(z) when $\upalpha$ ₁(z) + 2$\upalpha$ ₂ $U^{\prime }$(z) = c with c constant. Substituting this into Eq. (3.25) for the mean entropy production gives $I_{PCF} = { 2}\int {{\text{d}}z\upalpha_{ 1} \left( z \right)U^{\prime } \left( z \right) \propto \int {{\text{d}}zU^{\prime } \left( z \right)^{ 2} } }$ (Eq. 3.26), where we have dropped the constant term $c\int {{\text{d}}zU^{\prime } \left( z \right) \, = -cRe.}$ Thus effectively we can set $\upalpha$ ₁(z) ∝ $U^{\prime }$(z) in (3.25). A similar argument applies to plane Poiseuille flow, leading to the same expression for I _PPF (Eq. 3.30).