Comments on a derivation and application of the ‘maximum entropy production’ principle

We show that (1) an error invalidates the derivation (Dewar 2005 J. Phys. A: Math. Gen. 38 L371) of the maximum entropy production (MaxEP) principle for systems far from equilibrium, for which the constitutive relations are nonlinear; and (2) the claim (Dewar 2003 J. Phys. A: Math. Gen. 36 631) that the phenomenon of ‘self-organized criticality’ is a consequence of MaxEP for slowly driven systems is unjustified.


Systems far from equilibrium
Using the same assumptions and notation as Dewar [2], we have verified his results through equation (13) . . , f m ), where f i 's are functions whose known average values are (F 1 , . . . , F m ) and λ i is the Lagrange multiplier associated with F i in the maximization of the information entropy in the MaxEnt approach. Dewar notes that 'the generic FT [2, equation (12)] has important implications for the functional form of the 9718 Comment relationship between λ and F '. He then states, correctly, that the quadratic approximation to p(f ) in the vicinity of f = F is given by his equation (14) p However, there is no reason to expect, as Dewar requires in the line following equation (14), that the equation be valid for all f including f far from F , and therefore there is no basis for concluding (equation (15)) that λ k = j A jk (F )F j . Indeed, if it were true, equation (15) would imply (also using equation (10) for the definition of A nk (F )) that hence ∂A jk (F )/∂F n = 0 for arbitrary F . But this is only true when A is independent of F (i.e., linear constitutive relations), contrary to Dewar's claim. The point is that approximation (14) of [2] (our equation (1)) is typically valid only for f near F while the FT, equation (13), involves p at both ±f . However, since f and −f can both be near F only when F is near zero however, equations (13) and (14) hold simultaneously only for small F , i.e., in the linear (near-equilibrium) regime where A(F ) is independent of F .
Defining (as does Dewar) the mean dissipation D = d where d ≡ 2 k λ k f k and using the definition of A nk (F ), we find The orthogonality condition for D is essential to the subsequent derivations in [2] of the principles of 'maximum dissipation' and 'minimum dissipation', and hence of the corresponding principles of MaxEP and minimum entropy production, respectively. Contrary to the claim in [2], therefore, the derivations using this orthogonality condition do require 'the usual near-equilibrium assumption of linear constitutive relations' [2, p L375]. Thus the question of the existence of possible extremal principles (and in particular, of MaxEP) that might apply to far-from-equilibrium regimes (having nonlinear constitutive relations) has not been settled by [1,2].

Illustrative example
To illustrate these points explicitly, we make use of an instructive model proposed by Bruers in a recent preprint [3] that also discusses Dewar's work. We consider a single-spinchain simplification of that (two-spin-chain) model, make some small changes in definition (corresponding to differences in factors of τ , the number of spins in the chain), present some results that correspond to those in [3], and go on to show how and why Dewar's argument [2] fails. In the single-chain model, the distribution p i is over all sequences of spins i ≡ (σ 1 , σ 2 , . . . , σ τ ), where each σ t = ±c. We choose the quantity f to be the sum of spins f (i) ≡ σ 1 + · · · + σ τ , and the constraint [2, equation (2)] to be (Dewar's vectors λ, f and F , and matrix A are all scalars here since m = 1.) As Dewar requires (equation (11)), the possible sequences i can be grouped into 1 . In order that F → constant as τ is increased, c should go as 1/τ . We define c = 1/τ for convenience in what follows.
For this model we obtain, exactly (again as in [3], except for factors of τ and for one spin chain rather than two): the partition function For finite τ , only a discrete set of f values can occur; therefore, to calculate the p.d.f. p(f ) of f , we pass to the large-τ limit. Using Stirling's approximation for the factorial and dropping terms of O(ln τ/τ, 1/τ ) on the RHS, we obtain Expanding this about f = F , we obtain the quadratic approximation 1

However, [2, equation (15)] is violated: the LHS is λ(F ) = τ arctanhF , while the RHS is
. These two quantities are only equal in the limit F → 0, where both reduce to τ F . In this limit the constitutive relation between λ and F becomes linear, and the quantity A(F ) becomes independent of F. Dewar emphasizes that the constitutive relation (equation (15)) is nonlinear owing to the dependence of A on F ; however, as we see, equation (15) is only satisfied in the linear limit in which A becomes independent of F.
Bruers [3, p 25] notes that the orthogonality condition for D is only valid for linear systems, contrary to Dewar's [2] claim. However, the error in [2] that invalidates Dewar's claim is not identified in [3], and [3, equation (40)] repeats Dewar's assertion that D is equal to 2 k λ k f k and also to 2 j,k A jk (F )F j F k . As we have seen, the latter assertion is incorrect in the nonlinear (far-from-equilibrium) regime.

MaxEP and SOC
We turn now to Dewar's first article [1], in which it is argued that the occurrence of SOC in flux-driven systems follows generally from the principle of MaxEP, in the limit where the driving is infinitely slow with respect to the internal dynamics. This claim has great potential significance, since there is at present no clear, general understanding of the origins of SOC; nor have necessary and sufficient conditions for the occurrence of the phenomenon been delineated. We demonstrate, however, that the arguments about SOC in [1] are flawed, and that no conclusion about the occurrence of the phenomenon in the model considered there can be properly drawn.
To see this, consider the sandpile-like system considered in [1], using the same definitions and notation. In particular, F is the output flux of grains from the pile, F ext is the fixed external input of grains and p(F |F ext ) is the output-flux probability distribution for given F ext . Reference [1] starts with the assumption (in equation (26)) that where Z(F ext ) ≡ dF exp{H (F |F ext )} and (equation (27)) for small F, with the real constants r and g, respectively, positive and negative.
Dewar's claim that the model defined in this way exhibits SOC rests on the assertion that the quantity F 2 ≡ dF F 2 p(F |F ext ) diverges in the limit F ext → 0. The only divergence in the integral defining F 2 can come from large F, since the integrand is well behaved near F = 0. In particular, for F 2 to be infinite at a given F ext , p(F |F ext ) must not decrease more rapidly than F −3 as F → ∞, aside from slowly varying corrections such as logarithms. Such an algebraically slow decrease of p(F |F ext ) with F means that the distribution, D(n), of avalanche sizes n, also falls off algebraically slowly with n for large n, which is the defining characteristic of SOC [4].
The assertion that F 2 diverges is unjustified, however. In fact, since the behaviour of H (F |F ext ) at large F is unspecified in the model, no conclusion about the large-F behaviour of p(F |F ext ) and hence about the behaviour of F 2 can be drawn. If one assumes a particular large-F form for p(F |F ext ), moreover, that assumed form alone determines whether the model has SOC or not. If, for example, one assumes that p(F |F ext ) ∼ 1/F α for some power α as F → ∞, then one has by fiat defined the model to exhibit SOC. (Any value of α suffices to ensure SOC, even though F 2 is finite for α > 3.) If, on the other hand, one assumes that p(F |F ext ) decreases faster than a power of F-exponentially say-then by definition the model does not exhibit SOC.
It remains to understand, then, how [1] concludes that F 2 diverges as F ext → 0. The point is that F 2 is computed in [1] under a ('mean-field') approximation wherein H (F |F ext ) is expanded to quadratic order in (F − F ) (with F = F ext = √ −r/2g in mean-field approximation), the cubic and quartic terms being neglected. Implicitly, moreover, this quadratic form is assumed to hold not just for small F but for all F, so that F 2 can be computed 2 . Since the coefficient of the quadratic term vanishes like |g|F 2 ext (i.e., like r) as F ext → 0, the integral defining F 2 diverges in this limit. Note, however, that if one keeps the higher order terms in the expansion of H (F |F ext ), and again assumes that the resulting form-now quartic in (F − F )-holds for all F to allow the computation of F 2 , one readily sees that F 2 stays finite at F ext = 0, where one finds F 2 ∝ 1/ √ |g|. (This result follows immediately from changing the integration variable from F to y ≡ F |g| 1/4 in the integrals defining Z and F 2 , assuming that |g| > 0 at F ext = 0.) Thus the apparent divergence of F 2 is an artefact of using the mean-field approximation and assuming that the resulting quadratic form remains valid for large F. Though it has yet to be demonstrated, Dewar's conjecture in [1] that MaxEP or a similar principle underlies SOC remains an intuitively appealing notion.