Approximate embeddings in statistical mechanics

It is shown that any classical (possibly stochastic) dynamical system can be approximately embedded with arbitrarily high precision into a quantum system described in a sufficiently large, but finite dimensional Hilbert space. Two factors contribute to the number of Hilbert space dimensions needed in the construction. The description of the thermodynamic state space gives rise to a contribution of the order of the number of balls needed to cover the state space at the required accuracy. This factor grows exponentially with the system size. In contrast, the modelling of the dynamics increases the dimension only by a factor independent of the system size, and is hence negligible in the thermodynamic limit.