By Eric Bertin
This concise primer (based on lectures given at summer time colleges on complicated structures and on a masters measure path in advanced platforms modeling) will supply graduate scholars and novices to the sector with the elemental wisdom of the innovations and strategies of statistical physics and its strength for program to interdisciplinary topics.
Indeed, lately, statistical physics has began to draw the curiosity of a extensive group of researchers within the box of complicated procedure sciences, starting from biology to the social sciences, economics and desktop technological know-how. extra mostly, increasingly more graduate scholars and researchers believe the necessity to study a few uncomplicated innovations and questions originating in different disciplines with out unavoidably having to grasp the entire corresponding technicalities and jargon. in general talking, the pursuits 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 procedure thought of from the microscopic legislation ruling the dynamics of the person ‘entities’. those ambitions are, to some degree, additionally shared via what's these days referred to as ‘complex platforms technological know-how’ and for those purposes, platforms studied within the framework of statistical physics will be regarded as one of the easiest examples of advanced systems—allowing moreover a slightly good constructed mathematical remedy.
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Extra info for A Concise Introduction to the Statistical Physics of Complex Systems
If configurations are no longer discrete, but are defined by a continuous variable yt , a probability density T˜ (yt+1 |yt ) needs to be introduced, in such a way that T˜ (yt+1 |yt )dyt+1 is the probability to choose a new configuration in the interval [yt+1 , yt+1 +dyt+1 ], starting from a given configuration yt at time t. The equivalent of the normalization condition Eq. 139 now reads ∞ T˜ (yt+1 |yt )dyt+1 = 1. 140) −∞ Another generalization consists in replacing the discrete time steps by a continuous time evolution.
If x0 = 0 with probability 1, one has x02 = 0, and xt2 = t. This √ means that the typical position reached by the walk after t steps is of the order of t. The present random walk problem bears a direct relationship to the central limit theorem [2, 3]. The central limit theorem states that a sum of independent and identically distributed (iid) random variables, once conveniently rescaled to have zero mean N u i is and unit variance, converges to a Gaussian distribution. 161) converges to the normal distribution 1 2 P(z) = √ e−z /2 .
Hence the Dirac distribution is a tool which gives a mathematical meaning to the limit τcol → 0, and which thus simplifies formal calculations. In the above presentation, we chose to keep a small but finite τcol in order to give the reader a better intuition of the physics behind the calculations, and to avoid possible difficulties arising from the use of a new mathematical object. 2 Relation to Random Walks After having introduced the Langevin equation from a physical perspective (that of a probe particle immersed in a fluid), it is interesting to present the Langevin equation from another perspective, that of random walks.
A Concise Introduction to the Statistical Physics of Complex Systems by Eric Bertin