# Landau Theory of fluctuations

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.