Download e-book for iPad: A Concise Introduction to the Statistical Physics of Complex by Eric Bertin

By Eric Bertin

ISBN-10: 3642239226

ISBN-13: 9783642239229

ISBN-10: 3642239234

ISBN-13: 9783642239236

Introduction.- Equilibrium Systems.- Nonequlibrium Systems.- References

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Extra resources for A Concise Introduction to the Statistical Physics of Complex Systems

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164, one finds d Pn ν ν = −ν Pn (t) + Pn+1 (t) + Pn−1 (t). 166) The evolution of the probability distribution Pn (t) can be evaluated from Eq. 166, for instance by integrating it numerically. However, one may be interested in making analytical predictions in the large time limit, and such a discrete-space equation is not easy to handle in this case. 166) can be 40 2 Non-Stationary Dynamics and Stochastic Formalism approximated by a continuous space equation, namely, a partial differential equation.

Following standard mathematical methods, the general solution of Eq. , the noiseless equation) and of a particular solution of the full equation. The general solution of the homogeneous equation is vh (t) = A0 e−γ t , where A0 is an arbitrary constant. In order to determine a particular solution, one can use the so-called “variation of the constant” method, which indicates that such a solution should be searched in the form vp (t) = A(t)e−γ t , that is, simply replacing the constant A0 in the solution vh (t) of the homogeneous equation by a function A(t) to be determined.

201) where δk,k is the Kronecker delta symbol, equal to 1 if k = k and to zero otherwise. 202) t, ξ t ξt = a 2 ν δk,k . 203) This expression is the analog of Eq. 194, and the role played by τcol in the physical approach to the Langevin equation is now played by t. To give further evidence for this correspondence, we point out that δk,k / t can be interpreted as the discretized version of the Dirac distribution. 195) of the Dirac delta function. 205) where Q(x) ≡ a 2 νq(x), and where the noise ξ(t) satisfies ξ(t) = 0, ξ(t)ξ(t ) = δ(t − t ).

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


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