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We uncover a finite-time dynamical phase transition in the thermal relaxationof a mean-field magnetic model. The phase transition manifests itself as a cuspsingularity in the probability distribution of the magnetisation that forms ata critical time. The transition is due to a sudden switch in the dynamics,characterised by a dynamical order parameter. We derive a dynamical Landautheory for the transition that applies to a range of systems with scalar,parity-invariant order parameters. Close to criticalilty, our theory reveals anexact mapping between the dynamical and equilibrium phase transitions of themagnetic model, and implies critical exponents of mean-field type. We arguethat interactions between nearby saddle points, neglected at the mean-fieldlevel, may lead to critical, spatiotemporal fluctuations of the orderparameter, and thus give rise to novel, dynamical critical phenomena.

Finite-time dynamical phase transition in non-equilibrium relaxation