Journal of Statistical Physics, Vol. 26, No. 2, 1981 Fluctuations Around the Boltzmann Equation Herbert Spohn 1 Received January 12, 1981 For a system of hard spheres we prove the convergence of the second moment of the fluctuation field in the low-density limit. This extends a previous result by van Beijeren, Lanford, Lebowitz and Spohn(I) to nonequilibrium states. KEY WORDS: Hard sphere gas; Grad limit; fluctuation theory. 1. INTRODUCTION TO THE PROBLEM OF FLUCTUATIONS We consider a system of hard spheres of diameter c and unit mass inside a bounded region A c R 3. The hard spheres collide elastically amongst them- selves and are specularly reflected at the boundary of A. In essence this prescription defines the dynamics of hard spheres. We denote the corre- sponding flow by T~ acting on the grand canonical phase space F = U n>o(A x R3)n. If x E F stands for the initial positions and momenta of the particles, then T~x are the positions and momenta of the particles at the time t. We assume that the initial state of the system is given by a probability measure/~' on F which is absolutely continuous with respect to the Lebesgue measure. (In F there are configurations for which spheres overlap. Initially, and therefore at any other time, probability zero is assigned to these configurations. There are also other configurations lead- ing in the course of time to grazing and triple collisions. For these Tt" remains undefined. As to be discussed in the following section they form a set of measure zero with respect to/~'.) We want to understand the macroscopic behavior of the hard sphere gas at low density. In this regime it is natural to study the number n'(A, t) of particles in A at time t, where AcAxR 3 is some region of the I Theoretische Phys,, Universit/q.tMfinchen, Theresienstr. 37, 8 Mfinchen 2, Federal Republic of Germany. 285 0022-4715/81/1000-0285503.00/0 9 1981 Plenum Publishing Corporation