A Concise Introduction to the Statistical Physics of Complex by Eric Bertin

By Eric Bertin

This concise primer (based on lectures given at summer time colleges on advanced structures and on a masters measure path in advanced platforms modeling) will offer graduate scholars and beginners to the sector with the elemental wisdom of the ideas and strategies of statistical physics and its capability for software to interdisciplinary topics.

 Indeed, lately, statistical physics has began to draw the curiosity of a wide neighborhood of researchers within the box of complicated process sciences, starting from biology to the social sciences, economics and laptop technology. extra commonly, progressively more graduate scholars and researchers think the necessity to research a few uncomplicated suggestions and questions originating in different disciplines with out inevitably having to grasp all the corresponding technicalities and jargon. in general conversing, the targets of statistical physics will be summarized as follows: at the one hand to review platforms composed of a big variety of interacting ‘entities’, and at the different to foretell the macroscopic (or collective) habit of the method thought of from the microscopic legislation ruling the dynamics of the person ‘entities’. those objectives are, to some degree, additionally shared through what's these days referred to as ‘complex platforms technological know-how’ and for those purposes, platforms studied within the framework of statistical physics should be regarded as one of the easiest examples of complicated systems―allowing additionally a slightly good constructed mathematical therapy.

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154) C,C (C =C ) Summing Eqs. 154, and using the detailed balance property W (C |C) = W (C|C ), we obtain 1 dS = dt 2 ln P(C , t) − ln P(C, t) P(C , t) − P(C, t) W (C|C ). 155) As [ln P(C , t) − ln P(C, t)] and [P(C , t) − P(C, t)] have the same sign, one concludes that dS ≥ 0. 156) This is one possible statement, in the context of stochastic processes, of the second law of thermodynamics. Moreover, in the stationary state, d S/dt = 0, and one necessarily has for all pairs (C, C ) either Pst (C) = Pst (C ) or W (C|C ) = 0, where Pst (C) is the stationary probability distribution.

92) The non-zero solutions are m = ±m 0 with m0 = 3 √ Tc − T J −1 = 3 T T 1/2 . 93) It can be checked easily that the solution m = 0 corresponds in this case to a local maximum of f(m), and thus to a local minimum of P(m). Hence, there is a phase transition at T = Tc ≡ J , Tc being called the critical temperature. The most probable magnetization m 0 is called the “order parameter of the phase transition”, as the phase transition is characterized by the onset of a non-zero value of m 0 . In addition, the order parameter varies as m 0 ∼ (Tc − T )β for T close to Tc , with β = 1/2 here.

One considers a set of N independent and identically distributed random variables, drawn ∞ from a distribution p(x) such that −∞ x 2 p(x)d x is infinite. The cumulative distribx ution function is denoted as F(x) ≡ −∞ p(x )d x . 228) i=1 converges, for a suitable choice of the rescaling parameters a N and b N , to the Lévy distribution L(z; α, β) with 0 < α ≤ 2 and −1 ≤ β ≤ 1, if the following conditions are satisfied lim x→∞ F(−x) 1−β = 1 − F(x) 1+β 1 − F(x) + F(−x) = rα. 232) A few remarks are in order here: • The parameter α has the same interpretation here as in the example above, namely the tail of the distribution p(x) typically decays with an exponent 1 + α.

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