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 faculties on advanced structures and on a masters measure path in advanced structures modeling) will offer graduate scholars and rookies to the sphere with the elemental wisdom of the strategies and techniques of statistical physics and its strength for software to interdisciplinary topics.

Indeed, in recent times, statistical physics has began to draw the curiosity of a large group of researchers within the box of advanced procedure sciences, starting from biology to the social sciences, economics and machine technological know-how. extra normally, more and more graduate scholars and researchers think the necessity to research a few simple ideas and questions originating in different disciplines with out unavoidably having to grasp all the corresponding technicalities and jargon. usually talking, the ambitions of statistical physics could be summarized as follows: at the one hand to review structures composed of a big variety of interacting ‘entities’, and at the different to foretell the macroscopic (or collective) habit of the process thought of from the microscopic legislation ruling the dynamics of the person ‘entities’. those objectives are, to some degree, additionally shared by way of what's these days referred to as ‘complex structures technological know-how’ and for those purposes, structures studied within the framework of statistical physics will be regarded as one of the easiest examples of complicated systems—allowing additionally a slightly good built mathematical therapy.

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The non-zero, but small value of n(E) comes from the contribution to the average value of very rare samples, which include some configurations with exceptionally low (or high) energy. We can now evaluate the partition function of the REM, defined as 2N e−E α /T . 125) α=1 As all the energies E α are random variables, the partition function Z is also a random variable, which fluctuates from one realization of the disorder to another. 126) −ε0 with the notation n(ε) ˜ = N n(N ε). In Eq. 126, we have replaced n(ε) ˜ by n(ε) ˜ for |ε| < ε0 , and by 0 for |ε| > ε0 , consistently with the above discussion.

Beside the property ξ(t) = 0, its twotime correlation should be specified. Intuitively, one expects collisions occurring at different times to be essentially uncorrelated, so that one should have ξ(t)ξ(t ) = 0 42 2 Non-Stationary Dynamics and Stochastic Formalism for |t − t | τcol , where τcol is the typical duration of a collision. 175) where the function C(u) converges rapidly to zero when u → ∞. Altogether, Eq. 173 can be rewritten as: dv = −γ v + ξ(t). 176) Such an equation is called a linear Langevin equation.

4 A Simple Example: The One-Dimensional Random Walk A simple and illustrative example of stochastic process is the one-dimensional random walk, where a “particle” moves at random on a discretized line. Let us consider first the discrete time case: a particle can take only discrete positions x = . . , −2, −1, 0, 1, 2, . . on a line. Between times t and t + 1, the particle randomly jumps to one of the two neighboring sites, so that xt+1 = xt + t , with t = ±1 with equal probabilities. The random variables t and t , with t = t , are independent and identically distributed.

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