With respect to chapter 12 in the book Statistical Physics(Part 1) by Landau and Lifshitz, I am currently stuck at the intepretation of Fluctuation theory that Landau provides. In the neighbourhood of eqn. $119.6$ Landau mentions that when the system is in a state fairly close to equilibrium, then the fluctuations of the random variable $x^i$ about its equilibrium mean can be described(in the linear regime as)$$\dot{x^{i}} =- \lambda^{i}_{k}x^k$$ where the summation convention is implied and the $\lambda^{i}_{k}$'s being constant coefficients. From there Landau goes ahead to mention(in the neighbourhood of eqn $119.7$) that $$\dot\xi^{i} = -\lambda^{i}_{k}\xi^k $$ where the $\xi^i(t)$ defines (probably, even though i'm not sure) as the Conditional mean value of the random variable $x^i$ with the condition that it had the value $x^i$ at time $t_0 = 0$ i.e $$\xi^i(t) = \int dy^{i}P(y^i,t|x^i,0)y^i$$ My first doubt is whether this interpretation of $\xi^{i}$ as being the conditional probability is correct. Secondly Landau mentions that eqn $119.7$ directly follows from $119.6$. I do not quite understand why this is obvious.Thirdly, in the neighbourhood of eqn $119.7$, Landau mentions that the correlation functions themselves obey such a linearized equation in eqn $119.8$ I do not quite understand how he comes to such a conclusion. Any help on the explanation would really help me. At the same time, any references to literarture along the lines of this material in Landau would be really helpful.


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