Entropic Distance for Nonlinear Master Equation

More and more works deal with statistical systems far from equilibrium, dominated by unidirectional stochastic processes augmented by rare resets. We analyze the construction of the entropic distance measure appropriate for such dynamics. We demonstrate that a power-like nonlinearity in the state probability in the master equation naturally leads to the Tsallis (Havrda-Charv\'at, Acz\'el-Dar\'oczy) q-entropy formula in the context of seeking for the maximal entropy state at stationarity. A few possible applications of a certain simple and linear master equation to phenomena studied in statistical physics are listed at the end.


Definition and Properties of Entropic Distance
Entropic distance, more properly called "entropic divergence", is traditionally interpreted as a relative entropy, as a difference between entropies with a prior condition and without [1].It is also the Boltzmann-Shannon entropy of a distribution relative to another [2].Looking at this construction, however, from the viewpoint of a generalized entropy [3], the simple difference or logarithm of a ratio cannot be hold as a definition any more.
Instead, in this paper, we explore a reverse engineering concept: seeking for an entropic divergence formula at the first place, which is subject to some wanted properties, we consider entropy as a derived quantity.More precisely we seek for entropic divergence formulas appropriate to a given stochastic dynamics, shrinking during the approach to a stationary distribution, whenever it exists, and establish the entropy formula from this distance to the uniform distribution.By doing so we serve two goals: i) having constructed a non-negative entropic distance we derive an entropy formula which is maximal for the uniform distribution, and ii) we come as near as possible to the classical difference formula for the relative entropy.
We start our discussion by contrasting the definition of the metric distance, knwon from geometry, to the basic properties of an entropic distance.The metric distance possesses the following properties: 1. ρ(P, Q) ≥ 0 for a pair of points P and Q, The entropic divergence on the other hand is neither necessarily symmetric, nor can satisfy a triangle inequality.On the other hand it is subject to the second law of thermodynamics, distinguishing the time arrow from the past to the future.We require for a real functional, ρ[P, Q], depending on the distributions Pn and Qn, the followings to hold: The only constraint is to start with a core function, σ(ξ) with a definite concavity.Jensen inequality tells for σ ′′ > 0 that For satisfying property 1 and 2 one simply sets σ(1) = 0. Interesting enough, but this setting suffices also for the satisfaction of the second law of thermodynamics, formulated above as further constraints 3 and 4. As a consequence of the symmetrization it also follows that s(1) = 0 and s ′′ > 0.
s(ξ) ≥ 0. In this way the kernel function and hence each summand in the symmetrized entropic divergence formula is nonnegative, not only the total sum.
The antisymmetrized sum in the above equation is merely to ensure the conservation of the norm, The basic trick is to apply the splitting ξm = ξn + (ξm − ξn) to get Here the sum in the first term vanishes due to the very definition of the stationary distribution, Qn.For estimating the remaining term we utilize the Taylor series remainder theorem in the Lagrange form.We recall the Taylor expansion of the kernel function σ(ξ), with cmn ∈ [ξm, ξn].Here the first derivative term has occurred in eq.( 6).This construction delivers Here the first sum vanishes again: after exchanging the indices m and n in the first summand, the result is proportional to the total balance expression, which is zero for the stationary distribution.With positive transition rates, wnm > 0 the approach to stationary distribution, ρ ≤ 0 is hence proven for all σ ′′ > 0. We note that we never used the detailed balance condition for the transition rates, only the vanishing of the total balance, which defines the stationary distribution.This proof, without recalling the detailed balance condition as Boltzmann's famous H-theorem did, is quite general.Any core function with positive second derivative and the scaling trace form co-act to ensure the correct change in time.By using the traditional choice, σ(ξ) = − ln ξ, we have σ ′ = −1/ξ and σ ′′ (ξ) = 1/ξ 2 > 0, satisfying indeed all requirements.The integrated entropic divergence formula (no symmetrization) in this case is given as the Kullback-Leibler divergence : There is a rationale behind using the logarithm function.It is the only one being additive for the product form of its argument, mapping factorizing and hence statistically independent distributions to an additive entropic divergence kernel: For P n P (2) n also Q n therefore we have ξ n .
Finally we would like to treat this entropic divergence as an entropy difference.This is achieved when comparing the stationary distribution to the uniform distribution, Un = 1/W, n = 1, 2, . . .W .Using the above Kullback-Leibler divergence formula one easily derives with being the Boltzmann-Gibbs-Planck-Shannon entropy formula.From the Jensen inequality it follows ρ

Entropic divergence evolution for nonlinear master equations
Detailed balance is also not needed for a more general dynamics.We consider Markovian dynamics, with a master equation nonlinear in the distribution, Pn, as The stationarity condition defines The entropic distance formula is sought for in the trace form (but this time without the scaling assumption): In order to use the remainder theorem one has to identify This ensures ρ < 0 for any κ ′′ > 0 and P = Q.
We again would like to interpret this entropic divergence as entropy difference.The entropic divergence of the stationary distribution from the uniform distribution Un = 1/W, n = 1, 2, . . .W is given by: with ST being the Tsallis entropy formula: From the Jensen inequality it follows ρ i.e. the Tsallis entropy formula is also maximal for the uniform distribution.The factor W q−1 signifies non-extensivity, a dependence on the number of states in the relation between the entropic divergence and the relative Tsallis entropy.

Master equation for unidirectional growth and reset
With the particular choice of the transition rates, wnm = µmδn−1,m + γmδn,0, one describes a local growth process augmented with direct resetting transitions from any state to the ground state labelled by the index zero [8].The corresponding master equation Ṗn = µn−1Pn−1 − (µn + γn) Pn is terminated at n = 1 and the equation for the n = 0 state takes care of the normalization conservation: For the stationary distribution one obtains and Q0 has to be obtained from the normalization.Table 1 summarizes some well known probability density functions, PDF-s, which emerge as stationary distribution to this simplified stochastic dynamics upon different choices of the growth and reset rates µn and γn.In the continuous limit we obtain with the stationary distribution Table 1: Summary of rates and stationary PDF-s.Finally we derive a bound for the entropy production in the continuous model of unidirectional growth with resetting.
First we study the time evolution of the ratio, ξ(t, x) = P (x, t)/Q(x).Using P = ξQ we get from eq.( 25): Using the same eq.for stationary Q(x) and dividing by Q we obtain Now we turn to the evolution of the entropic divergence, With the symmetrized kernel, s(ξ) = σ div (ξ) + ξ σ div (1/ξ) ≥ 0, one gets using ∂s ∂t = −µ(x) ∂s ∂x the following distance evolution, considering the boundary condition ξ(t, 0) = 1 and s(1) = 0: We note that for the Kullback-Leibler divergence the following symmetrized kernel function has to be used: σ(ξ) = − ln ξ leads to s(ξ) = (ξ − 1) ln ξ and in this way ensures dρ dt ≤ 0. In order to obtain a lower bound for the speed of the approach to stationarity, we use again the Jensen inequality for s(ξ): with any arbitrary p(x) ≥ 0 satisfying p(x) dx = 1.For pour purpose we choose p(x) = γ(x)Q(x)/ γQ dx.This leads to the following result: Note that the controlling quantity is actually the expectation value of the resetting rate, p(x)ξ(x) dx = γP dx = γ t .Since s(ξ) reaches its minimum with the value zero only at the argument 1, the entropic divergence ρ(t) stops changing only if the stationary distribution is achieved.In all other cases it shrinks.

Summary
Summarizing, in this paper we have presented a construction strategy for the entropic distance formula, designed to shrink for a given wide class of stochastic dynamics.The very entropy formula was then derived from inspecting this distance between the uniform distribution and the stationary PDF of the corresponding master equation.In this way for linear master equations the well-known Kullback-Leibler definition arises, while for nonlinear dependence on the occupation probabilities one always arrives at an accordingly modified expression.In particular for a general power-like dependence the Tsallis q-entropy occurs as the "natural" relative entropy interpretation of the proper entropic divergence.In the continuous version of the growth and reset master equation, a dissipative probability flow supported with an inflow at the boundary, a lower bound was given for the shrinking speed of the symmetrized entropic divergence using the Jensen inequality.
To finish this paper we would like to make some remarks on real world applications of the above discussed mathematical treatment.Among possible applications of the growth and resetting model we mention the network degree distributions showing exponential behavior for constant rates and a Tsallis-Pareto distribution [9] (in the discrete version a Waring distribution [10,11]) for having a linear preference in the growth rate, µn = α(n + b).For high energy particle abundance (hadron multiplicity) distributions the negative binomial PDF is an excellent approximation [12], when both rates µ and γ are linear functions of the state label.For middle and small settlement size distributions a log-normal PDF arise, achievable with linear growth rate, µ(x) and a logarithmic reset rate, γ(x) ∼ ln x.Citations of scientific papers and Facebook shares and likes also follow a scaling Tsallis-Pareto distribution [13,14], characteristic to constant resetting and linear growth rates.While wealth seems to be distributed according to a Pareto-law tail, the middle class incomes rather show a gamma distribution, stemming from linear reset and growth rates.For a review of such applications see our forthcoming work.

1 .
ρ[P, Q] ≥ 0 for a pair of distributions Pn and Qn, 2. ρ[P, Q] = 0 only if the distributions coincide Pn = Qn, 3. d dt ρ[P, Q] ≤ 0 if Qn is the stationary distribution, 4. d dt ρ[P, Q] = 0 only for Pn = Qn, i.e. the stationary distribution is unique.Although this definition is not symmetric in the handling of the normalized distributions Pn and Qn, it is an easy task to consider the symmetrized version, s[P, Q] ≡ ρ[P, Q] + ρ[Q, P ].This symmetrized entropic divergence inherits some properties from the fiducial construction.Considering a scaling trace form entropic divergence, ρ[P, Q] = n σ(ξn) Qn with ξn = Pn/Qn, to begin with, we identify the following symmetrized kernel function:

2
Entropic distance evolution due to linear stochastic dynamicsNow we study properties 3 and 4, by evaluating the rate of change of the entropic divergence in time.This change is based on the dynamics (time evolution) of the evolving distribution, Pn(t), while the targeted stationary distribution, Qn is by definition time independent.First we consider a class of stochastic evolutions governed by differential equations for Ṗn(t) ≡ dPn dt , linear in the distribution, Pn(t)[4].We consider the trace form ρ[P, Q]