A Minimal Model Shows an Internal Clock Slowing Down as Inputs Repeat

Physicists often treat time as something imposed from outside, a parameter that ticks along in the background while a system evolves. A new theoretical study flips that perspective by asking what happens when the clock itself is a property of the system, generated by the very inputs it receives. In a paper published in Discover Physics, Tatsuaki Tsuruyama formulates a minimal statistical-mechanical model in which a driven response system builds its own internal time from the distinct types of inputs, or codes, that arrive from the environment. The central result is strikingly simple: because repeated inputs do not create new states, the internal clock slows down monotonically and, when the number of possible code types is finite, it eventually saturates altogether, advancing asymptotically negligibly relative to physical time.

The model is deliberately spare. External inputs arrive at random times, described as a Poisson process with rate lambda, and each arrival carries one of a finite set of K possible codes, drawn independently from a fixed distribution. The system keeps only a coarse-grained record: a list of which code types have been observed at least once. Internal time is defined operationally as the number of distinct code types in that list, an integer-valued stochastic process that the author calls the occupancy count of the input history. Crucially, this is not a count of how many inputs have arrived. Even after many arrivals, if the same code keeps appearing, internal time barely moves, and that difference between arrival count and distinct-type count is the direct cause of the slowdown derived in the paper.

Under Poisson arrivals, the mathematics becomes exactly tractable through Poisson splitting, which shows that the arrivals carrying each code type form independent Poisson processes. The probability that a given code has been observed at least once by time t is one minus an exponential, and the mean internal time is the sum of these observation probabilities over all code types. Differentiating gives the mean clock rate, and each term of that derivative is proportional to the probability that the corresponding code remains unobserved. As time proceeds and the list of observed types grows, fewer unobserved codes remain, so the advancing speed decreases monotonically toward zero. Meanwhile the mean internal time itself is bounded above by K and approaches K as time goes to infinity, which means the ratio of mean internal time to physical time vanishes in the long-time limit.

The uniform case, in which every code type is equally likely, admits a closed-form expression in elementary functions. The mean internal time equals K times one minus an exponential, and the natural time scale is K divided by lambda: the larger the repertoire of possible codes, the longer physical time is needed before saturation, while a higher arrival rate brings saturation earlier. Inverting the mean relation yields the physical time required to reach a given mean internal-time level, and this inverse mapping diverges as the level approaches K. The physical-time increment needed for one additional unit of mean internal time grows sharply in later stages, a concrete formalization of the claim that internal time advances asymptotically negligibly relative to physical time. The author emphasizes that this is not a metaphor but a rigorous consequence of the boundedness of the mean.

The model also predicts how strongly internal time fluctuates around its mean. Because the observation indicators for different codes are independent Bernoulli variables under Poisson arrivals, the variance of internal time is a simple sum, bounded above by K divided by four, and it vanishes both at early times and in the infinite-time limit. In the uniform case the variance is unimodal, peaking at an intermediate stage, at a time equal to the natural scale times the logarithm of two. Fluctuations are therefore smallest when the clock has just started or has nearly run out of new states, and largest in the middle of its history, when roughly half the code types remain unobserved.

The framework extends naturally to a weighted notion of internal time. Assigning a positive description length to each code and summing the weights of all observed types gives a quantity that advances depending on which codes were seen, not merely how many. A coding-consistent choice of weights, based on the ideal Shannon relation between self-information and code length, makes rarer codes carry larger weights, so the weighted clock can still jump substantially late in the system’s history when a rare code finally appears. Under Poisson arrivals, this weighted internal time also slows down in the mean and saturates for finite K, but for heavy-tailed input its later-stage tail can be emphasized more strongly than that of the unweighted count.

Heavy-tailed input distributions receive particular attention as a mechanism for delayed saturation. Using a truncated Zipf law, in which code frequency falls off as a power of rank, the author shows that many rare codes remain unobserved for long periods, so the mean internal time rises quickly at first but then leaves a long tail toward saturation. In an intermediate regime, the growth can be estimated as a sublinear, power-like increase, although the author cautions that this scaling should be read only as a qualitative guide outside that regime and near finite-size endpoints. As the power-law exponent increases, probability concentrates on the top few codes, making the early rise faster but the later tail even longer. Over observation windows shorter than the saturation scale, rare unobserved types may keep appearing, producing the empirical impression that new types never run out.

As an ancillary quantity, the study introduces conditional entropy to measure what is lost when only the internal clock is retained rather than the full list of observed codes. An identity derived from the chain rule of entropy shows that the difference between keeping the full list and keeping only the count equals exactly the uncertainty that remains in the list when the count is known. In the uniform case, all candidate lists consistent with a given count are equally likely, yielding a closed form and a unique maximization time at the same intermediate stage where the variance peaks. For non-uniform distributions, the conditional entropy obeys a universal upper bound, and the gap from that bound can be written as a mean Kullback-Leibler divergence, which is positive at non-trivial times and vanishes only when the count is concentrated at its extremes. Under Poisson arrivals, the uniform distribution is the unique achiever of the upper bound.

The author is careful to position the work. The model is explicitly distinguished from psychological models of subjective duration: internal time here is a state variable of a physical response system, defined by a reproducible update rule and analyzed as a stochastic process, not a reparameterization of physical time. The mathematical structure connects to occupancy problems, coupon-collector models, vocabulary growth, Heaps’ law and Zipf’s law, but the novelty lies in interpreting the occupancy count as the clock of a response system whose rate is determined by input statistics and the finite supply of unobserved states. The paper also notes a possible implication for biological response systems: as a system accumulates input history, the fraction of already observed types grows, so the same amount of physical time may correspond to fewer effective updates of the internal state, offering a minimal information-statistical perspective on why time may appear compressed. The author stresses that this is a possible implication of the model, not a complete account of psychological time perception, which also depends on attention, memory, emotion and behavioral structure.

Limitations are stated plainly. The minimal model retains no frequencies, order or inter-arrival intervals, the arrival process is restricted to Poisson for analysis, and no correspondence with external psychological or behavioral indices is assumed. The analysis focuses mainly on mean dynamics, although saturation also occurs at the sample-path level for finite K with positive code probabilities. Natural extensions include non-Poisson arrival processes such as periodic or self-exciting inputs, internal states that carry frequency or recent-history information, and heavy-tail limits in which the number of code types grows without bound and saturation is effectively never observed. Within its scope, however, the conclusion is unambiguous: the tick rate of an emergent internal clock is not imposed a priori but follows from the arrival rate, the code distribution and the progressive depletion of still-available state updates, so that on long time scales physical time can continue to advance while the system’s own clock barely moves.

Subject of Research: A statistical-mechanical model of an emergent internal clock in driven response systems

Article Title: Statistical mechanics of an emergent internal clock in driven response systems

Article References: Tsuruyama, T. (2026). Statistical mechanics of an emergent internal clock in driven response systems. Discover Physics, 2(1), Article 15. https://doi.org/10.1007/s44418-026-00014-y

Image Credits: AI Generated

DOI: 10.1007/s44418-026-00014-y

Keywords: statistical mechanics, internal time, stochastic processes, Poisson arrivals, occupancy problems, coupon collector, Zipf's law, conditional entropy, coarse graining, nonequilibrium physics, information theory, saturation