Stochastic Dynamics and Irreversibility (Graduate Texts in Physics)
Tânia Tomé, Mário J. de Oliveira
Language: English
Pages: 402
ISBN: B010WEQBJW
Format: PDF / Kindle (mobi) / ePub
This textbook presents an exposition of stochastic dynamics and irreversibility. It comprises the principles of probability theory and the stochastic dynamics in continuous spaces, described by Langevin and Fokker-Planck equations, and in discrete spaces, described by Markov chains and master equations. Special concern is given to the study of irreversibility, both in systems that evolve to equilibrium and in nonequilibrium stationary states. Attention is also given to the study of models displaying phase transitions and critical phenomena both in thermodynamic equilibrium and out of equilibrium.
These models include the linear Glauber model, the Glauber-Ising model, lattice models with absorbing states such as the contact process and those used in population dynamic and spreading of epidemic, probabilistic cellular automata, reaction-diffusion processes, random sequential adsorption and dynamic percolation. A stochastic approach to chemical reaction is also presented.The textbook is intended for students of physics and chemistry and for those interested in stochastic dynamics.
It provides, by means of examples and problems, a comprehensive and detailed explanation of the theory and its applications.
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distribution can be obtained if we consider the following linear combination, (2.113) where β > 0, which reduces to the form (2.112) when β = 1∕2. Given the distribution ρ(ξ j ), which we consider to be the same for all variables ξ j , we wish to obtain the distribution ρ ∗(x) of x. Denoting by g ∗(k) the characteristic function associated to the variable x, then (2.114) so that (2.115) where g(k) is the characteristic function associated to the variables ξ i . Assuming that the
a particle is (4.5) where the first term in the right-hand side is the friction force, proportional to the velocity; the second is an external force and the third is a random force. When the mass is very small and the friction is very large, we can neglect the term at left and write (4.6) Dividing both sides of these equation by the friction coefficient α, we get Eq. ( 4.1). Thus, the quantity f(x) in Eq. ( 4.1) is interpreted as the ratio between the external force and the friction
(5.86) where Z is a normalization constant, which is equivalent to (5.87) where U(x) = mV (x) is the potential associated to the force F(x), that is, , and Γ is related to the temperature by (5.88) and therefore B = 2α k B T. This last result and the result (5.87) allow us to interpret (5.65) as the equation that describes a particle subject to a force F(x) and in contact with a heat reservoir at temperature T. Both forces, −α v and F, describe the coupling to the heat reservoir. This
λ = λ c and λ > λ c . (b) Probability density ρ(x) versus x according to (9.79) for the same values of λ. The maxima of ρ correspond to the minima of F(x) To analyze the behavior of the function F(x), we determined its derivatives, given by (9.81) (9.82) The function F(x) has a single maximum, which we denote by x 0. For λ > 1, it is obtained from F′(x 0) = 0 and gives us . Notice that F″(x 0) = λ > 0 and therefore x 0 corresponds indeed to a minimum. When λ ≤ 1, the function F(x) is an
in favor or against it. The opinion of an individual changes with time according to certain rules that involve the opinion of the neighbors. At each time step an individual chooses at random one of his neighbors. If the neighbor’s opinion is different from his, the opinion is accepted with a certain probability to be specified later. If the opinions are the same, the individual’s opinion does not change. According to this rule, if all individuals of the community are favorable, there will be no
