By Eric Bertin
Introduction.- Equilibrium Systems.- Nonequlibrium Systems.- References
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Additional info for A Concise Introduction to the Statistical Physics of Complex Systems
2 Langevin and Fokker–Planck Equations 43 follows—since we look for a particular solution at this stage, there is no need to add a constant term to Eq. 180. Altogether, one finds for v(t), taking into account the initial condition v(0) = v0 , t v(t) = v0 e −γ t +e −γ t eγ t ξ(t )dt . 181) 0 Computing v(t)2 yields ⎛ v(t) = 2 v02 e−2γ t +e −2γ t ⎞2 t ⎝ e t ξ(t )dt ⎠ + 2v0 e γt −2γ t 0 eγ t ξ(t )dt . 182) 0 Now taking an ensemble average, the last term vanishes because ξ(t) = 0, and we get ⎛ ⎞2 t v(t)2 = v02 e−2γ t + e−2γ t ⎝ eγ t ξ(t )dt ⎠ .
Using −W (C |C)P(C, t) + W (C|C )P(C , t) ln P(C, t) C (=C) C ln P(C, t) W (C |C)P(C, t) − W (C|C )P(C , t) . 153) C,C (C =C ) Exchanging the notations C and C in the last equation, we also have dS = dt ln P(C , t) W (C|C )P(C , t) − W (C |C)P(C, t) . 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.
The interaction between magnetic atoms (which have to be described in the framework of quantum mechanics) is mediated by the non-magnetic atoms, and acquires an oscillatory behavior, depending on the distance ri j between the two spins: J (ri j )si s j . 122) i, j The interaction constant J (ri j ) is a given function of the distance ri j , which oscillates around 0, thus taking both positive and negative values. The amplitude of the oscillations decays as a power-law of the distance. As the distances between atoms are random, the interactions between atoms have a random sign, which is the basic property of spin-glasses.
A Concise Introduction to the Statistical Physics of Complex Systems by Eric Bertin