  The mean-field dynamics of a Bose gas is shown to break down at time $\tau_h
= (c_1/\gamma) \ln N$ where $\gamma$ is the Lyapunov exponent of the mean-field
theory, $N$ is the number of bosons, and $c_1$ is a system-dependent constant.
The breakdown time $\tau_h$ is essentially the Ehrenfest time that
characterizes the breakdown of the correspondence between classical and quantum
dynamics. This breakdown can be well described by the quantum fidelity defined
for reduced density matrices. Our results are obtained with the formalism in
particle-number phase space and are illustrated with a triple-well model. The
logarithmic quantum-classical correspondence time may be verified
experimentally with Bose-Einstein condensates.
