A partition function representation through Grassmann variables

We propose a formula for a classical partition function $Z_N$ that does not involve the Hamilton function of the system. In the general case, we avoid passing to canonical variables $(\mathbf p,\mathbf x)$ at the price of extending the space of Lagrange variables $(\mathbf v,\mathbf x)$ by introducing “additional velocities” $\bar{\mathbf u},\mathbf u$, which are the generators of a Grassmann algebra. In this space, the partition function $Z_N$ is the integral of a Gibbs-type distribution, whose explicit form is determined by the system Lagrange function. We calculate the partition function of a model system governed by the Darwin Lagrange function.