Dynamics of small fluctuations in Boltzmann kinetics

For clarity, we refer to Boltzmann kinetics as the theory of non-equilibrium states of a system of identical weakly interacting particles (the Hamiltonian of interaction H^1 ⁓ λ,λ<<1). The minimum set of parameters η1 that describes the state of such a system is assumed to be the set of occupancy numbers np of single-particle states (η1≡{np}, p is a state number). The presence of additional parameters (their set is denoted by η2) that describe the state of the system indicates, according to Onsager, the presence of fluctuations in the system. We discuss in detail the fluctuations of the squares of the occupation numbers mp (η2≡{mp}). The paper uses the Bogolyubov method of reduced description of non-equilibrium states with the Peletminskii–Yatsenko scheme application. Statistical operators ρ0(η1) and ρ(η1,η2) describe, respectively, the states of the system without fluctuations and with fluctuations and are expressed by perturbation theory in λ through quasi-equilibrium statistical operators ρq0(η1) and ρq(η1,η2). It can be argued that the description of the system by the statistical operator ρ(η1,η2), i.e., the states with fluctuations, is realized in longer evolution times than the description of the states without fluctuations. The average values of the observed quantities can be calculated in the state without fluctuations ρ0(η1) by the perturbation theory in the interaction parameter λ, using Wick rules for ρq0(η1). For a statistical operator, such an operation is impossible, since for the quasi-equilibrium statistical operator ρq(η1,η2) Wick-type rules do not exist (this operator is an exponential function of the operator form of the 4th power of secondary quantization operators). To overcome this problem, we develop an expansion for ρq(η1,η2) in terms of small powers of deviation δη2=η2−η20 (δη2 ⁓ μ,μ<<1), where η20 is the average value of η2 in the state ρq0(η1). The quantity δη2 characterizes the magnitude of the fluctuation in Boltzmann kinetics. The introduction of the new small parameter is similar to the method of correlation function calculation proposed in our previous work for spin systems (Low Temp. Phys.51, 323 (2025) [Fiz. Nyzk. Temp.51, 357 (2025)]).