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(generally growing) maintaining its basic statistical prop- erties stable [5, 7, 15, 24]. Using eq. (5), condition (6) is replaced by: 

We can replace eq. (9) alternatively by the following statement: For any ϵ > 0 there exists n ∈ N such that, for any n ′ > n : 

The main objective of the paper is to find the expected distribution p$_{n}$ ( s$_{i}$ ) consistent with eq. (10). The case µ = 0 would correspond to systems where, although growing in size, its complexity (and thus, its statistical entropy) is bounded or grows sublinearily with log n , a case studied in [51]. Here, we are interested in the in- termediate case, where µ ∈ (0 , 1). This characterization would depict systems with some balance among ordering and disordering forces, and thereby displaying a dissipa- tion of statistical entropy proportional to the maximum entropy achievable for the system in equilibrium. There- fore, we will refer to the problem of finding solutions for eq. (9) as the entropy restriction problem . A computa- tional test for this result can be illustrated by the model results shown in fig.(1). The picture shows a spatial snap- shot of the local population densities of a model of urban growth displaying Zipf’s law [23]. The normalized en- tropy evolves towards a stationary value µ ≈ 0 . 65 consis- tently with our discussion. This is true in spite that this model exhibits wide fluctuations due to its intermittent stochastic dynamics. 

## III. EMERGENCE OF ZIPF’S LAW IN STOCHASTIC SYSTEMS 

As pointed out in [36], the main difficulty we face in this kind of equations is that we are not dealing with an extremal problem, since our value of entropy is pre- viously fixed and it is neither minimum nor maximum, in Jaynes’ sense [60]. Thus, classical variational meth- ods, which have been widely used with great success in statistical mechanics [60–63], do not apply to our prob- lem -although recently it has been shown that variational approaches using Fisher information and physically rele- vant constraints lead to Power-laws whose exponent can be close to 1 [64]. We also must take into account that the particular properties of Zipf’s law create an addi- tional difficulty if the studied systems display, a priori, an unbounded number of possible states. Specifically, we refer to the non-existence of finite moments and normal- ization constant in the thermodynamical limit. However, as we shall see, these apparently undesirable properties will be the key to our derivation. 

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## A. Properties of the entropies of a power law 

Let us briefly summarize the properties of the entropies of power-law distributed systems, which will be used to derive the main results of this work (For details, see Ap- pendix A). Such properties are intimately linked with the behavior of the Riemann Zeta function , ζ ( γ ) [65]: 

In the real line, this function is defined in the interval γ ∈ (1 , ∞ ), displaying a singularity for γ → 1 $^{+}$. 

Now, let us suppose that the system contains n states and the probability to find the i -th most likely states decay as a power-law, i.e., p$_{n}$ ( s$_{i}$ ) ∝ i − $^{γ}$. For the sake of simplicity, we will refer to its associated entropy as H ( n, γ ) and to its normalized counterpart as h ( n, γ ), i.e.: 

The most basic properties concern the global behavior of H ( n, γ ). It is straightforward to check that H ( n, γ ) 

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