The purpose of this paper is to study the problem of deriving the Landau equation as a limit for large \(N\) of an \(N\)-particle system. The Landau equation under consideration is \[ (\partial_t + v\cdot \nabla_x)f = Q_L(f,f), \] where \(f(x,v,t\) is the one particle distribution, \(x \in \mathbb R^3,\; v \in \mathbb R^3\), and \(t \geq 0\) is time. The operator \(Q_L(f,f)\) is the Landau collision operator for Coulomb interaction. The paper begins an analysis and obtains some preliminary results on the weak-coupling limit for a Hamiltonian particle system.
It is assumed that the number of particles, \(N\), is very large, \(N=1/\epsilon^3\), and the interaction strength rather moderate. After a scaling in terms of \(\epsilon\) the particle system is given by \[ \frac{d}{dt} x_i=v_i,\;\; \frac{d}{dt} v_i = -\frac{1}{\sqrt{\epsilon}} \sum_{^{j=1,N}_{j \neq i}} \nabla \phi(\frac{x_i-x_j}{\epsilon}), \] where \(\phi\) is a smooth two-body interaction potential. To obtain a statistical description of the system (1) let \({\mathbf X}_N = \{x_1,\ldots,x_N \},\; {\mathbf V}_N = \{v_1,\ldots,v_N \},\; x_i \in \mathbb R^3,\; v_i \in \mathbb R^3\), and \({\mathbf W}^N = {\mathbf W}^N({\mathbf X}_N,{\mathbf V}_N)\) be a symmetric probability distribution. The weak-coupling limit is obtained when \(\epsilon \rightarrow 0\), and this limit is investigated by introducing the BBKG hierarchy for the j-particle distribution denoted \(f_j^N(X_j,V_j)\) which is derived from \({\mathbf W}^N({\mathbf X}_N,{\mathbf V}_N)\). The initial value \(f_j^0\) is introduced, and the equation for \(f_j^N(t)\) is written in terms of a perturbation of the free flow \(S(t)\) defined as \((S(t)f_j)(X_j,V_j) = f_j(X_j-V_jt,V_j)\). The conjecture is made that \(f_j^N\) can be expressed as \(f_j^N = g_j^N + \gamma_j^N\) where \(g_j^N\) is the main part of \(f_j^N\) and \(\gamma_j^N\) is small and strongly oscillating. The quantities \(g_j^N\) and \(\gamma_j^N\) satisfy a coupled hierarchy of partial differential equations in which \(g_j^N,\gamma_j^N\) depend on \(g_{j+1}^N,\gamma_{j+1}^N\). By eliminating the function \(\gamma_j^N\) in these equations the authors arrive at a closed hierarchy of integro-differential equations for \(g_j^N\). These equations are written symbolically as \(g_j = S(t)f_j^0 + A_{j+1}g_{j+1}\), and are solved by an iterative scheme \( g_j^0 = S(t)f_j^0,\;\; g_j^{(n+1)} = S(t)f_j^0 + A_{j+1}g_{j+1}^{(n)}, \; n=0,1,\ldots. \) The goal is to show that \(g_1^{(1)}(t) = \tilde g_1^N(t)\) is consistent with the Landau equation. This result is proved in a theorem that states that given some conditions on \(f_0\) and the potential \(\phi\) then \[ \lim_{\epsilon \rightarrow 0} \tilde g_1^N(t) = S(t)f_0 + \int_0^t d\tau S(t-\tau) Q_L(S(\tau)f_0, S(\tau)f_0), \] and \(\lim_{\epsilon \rightarrow 0} \tilde\gamma_1^N = 0\). Thus in the weak-coupling limit the solution derived from the particle system shows a consistency with the corresponding solution to the Landau equation obtained as a perturbation of the free flow. The last section of this paper extends the result of this theorem to the \(j\)-marginal distribution and shows the propagation of chaos at the first order in time.