3 (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. 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. 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 ) h ( n, γ ) = 1 log n ( γ Z n ∑ i =1 log i i γ + log Z ) . (12) ζ ( γ ) = ∞ ∑ k =1 1 k γ . (11) | h ( n $^{′}$) − µ | < ϵ. (10) lim n $_{→∞}$h ( n ) = µ (9) FIG. 1: (Color online) An example of the behavior of the normalized entropy for a multiplicative stochastic process ex- hibiting Zipf’s law. Here we use the model described in [23] using a 80 × 80 lattice where each node is described by a den- sity of population ρ ( i, j ). The rules of the model are very simple: i) At every time step, each node loses a fraction α of its contents, which is distributed among its four nearest neighbors. ii) At time t + 1 the local population is multi- plied, with probability p , by a factor p − $^{1}$. Furthermore, with probability 1 − p , the population of a node is set to zero. Ad- ditionally, at each step a random number η is added to every node. In this way, we avoid falling into an absorbing state ρ = 0. Here we use 0 < η < 0 . 01, α = 1 / 4 and p = 3 / 4. This is an extremly simplified (and yet successful) model of urban population dynamics. A snapshot (for t = 500) is shown in (a) where we can appreciate the wide range of local densities, following Zipf’s law (b). If we plot the evolution of the nor- malized entropy µ over time (averaged over 10 2 replicas) we observe a convergence towards a stationary value µ ≈ 0 . 65. A. Properties of the entropies of a power law III. EMERGENCE OF ZIPF’S LAW IN STOCHASTIC SYSTEMS 10 0 10 1 10 2 10 3 10 4 city rank 10 -6 10 -4 10 -2 10 0 Local density 0 100 200 300 400 500 time 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.0 b c a