Fluctuation Dissipation Theorem
The Fluctuation Dissipation Theorem (FDT) appears in various forms in non-equilibrium statistical physics. The most well-known form is the Einstein Relation, which describes the relationship between the diffusion coefficient D and friction coefficient γ for a particle moving under fluid friction and random forces, without other external forces: \begin{equation} D \gamma = k_B T \end{equation} This relationship between D, representing the magnitude of particle motion fluctuations, and γ, related to dissipation, gives rise to the name Fluctuation Dissipation Theorem. I aim to review and summarize R. Kubo’s well-organized lecture notes on this topic.
Introduction
The motion of colloidal particles suspended in fluid represents a classic example of Brownian motion. Such particles experience two main effects from fluid molecule collisions: friction force and random force. Since both effects arise from the same underlying cause, the friction force and random force must maintain a specific relationship, which we call the fluctuation-dissipation theorem.
FDT is generally expressed as a relationship between the response to external stimuli and inherent fluctuations in the absence of stimuli. The response is typically characterized by response functions, susceptibility, or impedance, while fluctuations are given by correlation functions of physical quantities in thermal equilibrium, or their spectra. Consequently, FDT appears either as predictions of fluctuations from impedance (Nyquist theorem) or as relationships between fluctuations and susceptibility (Onsager’s symmetry).
Despite its fundamental importance, FDT received relatively late recognition. This delay is understandable, as the Boltzmann equation was long considered the primary tool in non-equilibrium statistical physics. Although Einstein, Nyquist, and Onsager conducted related research in the early 1900s, comprehensive studies only emerged in the 1950s through works by Callen and Welton (1951), Callen and Greene (1952), Takahashi (1952), and Kubo and Tomita (1954).
Einstein relation
The Einstein relation serves as the most frequently cited example of FDT. The drift velocity ud created by a potential field V is given by: \begin{equation} u_d = - \frac{1}{\gamma} \dv{V}{x} \end{equation} When including the diffusion current arising from density distribution ρ(x) changes, the total fluid flow becomes: \begin{equation} j(x) = - D\pdv{f(x)}{x} + u_d f(x) \end{equation} By combining two equilibrium conditions - the density function must be proportional to the Boltzmann factor f(x) ∝ exp(-V/kBT), and the net flow must vanish - we can derive: \begin{equation} D \gamma = k_B T \end{equation}
To express this in modern FDT form, let’s revisit the definition of the diffusion coefficient: \begin{equation} D = \lim_{t \rightarrow \infty} \frac{1}{2t} \expval{\qty{x(t) - x(0)}^2} \end{equation} Here, the average ⟨·⟩ represents the ensemble average under thermal equilibrium. This quantity describes the average spread over long time periods. Expressing displacement in terms of velocity: \begin{equation} x(t) - x(0) = \int_{0}^t u(t’) \dd t’ \end{equation} Substituting this into the diffusion coefficient definition yields the velocity autocorrelation function: \begin{align} D &= \lim_{t \rightarrow \infty} \frac{1}{2t} \int_0^t \dd t_1 \int_0^t \dd t_2 \expval{u(t_1)u(t_2)} \notag \newline &= \lim_{t \rightarrow \infty} \frac{1}{t} \int_0^t \dd t_1 \int_0^{t - t_1} \dd t_2 \expval{u(t_1)u(t’)} \newline &= \int_0^\infty \expval{u(t_0) u(t_0 + t)}\dd t \notag \end{align} From this perspective, we can confirm that the Einstein relation demonstrates how the friction coefficient γ relates to Brownian motion fluctuations, exemplifying the fluctuation-dissipation theorem.
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