Statistical mechanics of an emergent internal clock in driven response systems

We formulate a minimal statistical-mechanical model of an emergent clock variable in a driven physical response system. The system is driven by stochastic input arrivals in physical time and updates a coarse-grained internal state when a previously unobserved code type is incorporated. Internal time is defined as the number of distinct code types observed up to physical time. Under Poisson arrivals, the mean internal time is obtained by summing the observation probabilities of all code types, and the corresponding mean clock rate decreases monotonically because new coarse-grained state updates become progressively less likely. For finitely many code types with positive probabilities, the mean internal time saturates at the number of available code types, so internal time advances asymptotically negligibly relative to physical time. In the uniform case, we derive a closed-form expression in elementary functions and an inverse relation at the level of the mean, showing that the physical time required for one additional mean internal-time step grows sharply near saturation. We also derive the variance, introduce a weighted internal time based on code-dependent description lengths, and analyze heavy-tailed input as a mechanism for delayed saturation and long tails. Conditional entropy is used only as an ancillary measure of the information lost by retaining the internal-clock variable alone. Thus, internal time is treated as an emergent coarse-grained clock variable generated by stochastic driving, rather than as an externally imposed time parameter.