Entropic Time: Clock Defined by Entropy

• Entropic Time is a framework where time is derived from entropy production, information acquisition, or statistical state changes instead of a fixed external parameter.
• Various formulations employ thermodynamic, inferential, and geometric methodologies—using Boltzmann’s relation, Bayesian updates, and metrics like Fisher–Rao—to construct a temporal parameter.
• This approach has practical implications in modeling irreversible processes in thermodynamics, quantum dynamics, and cosmology, as well as in experimental and information-theoretic applications.

Searching arXiv for the cited papers and closely related work on entropic time. Entropic time denotes a family of constructions in which temporal order, duration, or clock rate is defined by entropy, entropy production, information acquisition, or fluctuation scales rather than by an externally postulated uniform parameter. The term is not used uniformly across the literature. In nonequilibrium thermodynamics it is introduced as a measure of irreversible change; in Entropic Dynamics it is a bookkeeping parameter ordering successive probability updates; in cosmological and geometric models it is tied to horizon entropy or to entropy along thermodynamic geodesics; and in recent quantum-information and analog-gravity settings it appears as a relational time defined from entropy exchange or accessible information (Martyushev et al., 2016, Caticha, 2010, Clementine et al., 2017, Barontini, 9 Sep 2025).

1. Conceptual domain and basic definitions

A central feature of the subject is that “time” is not treated as a single primitive across all formulations. In the thermodynamic construction of Martyushev and Shaiapin, one starts from Boltzmann’s relation

$S = k_B \ln \Omega$

and, for an isolated system undergoing irreversible processes only, defines an intrinsic time by

$d\tau = dS_{\mathrm{ir}}/N,$

so that the clock advances only when irreversible entropy is produced. In this form, entropic time is explicitly tied to the thermodynamic arrow of time (Martyushev et al., 2016).

In Entropic Dynamics, by contrast, entropic time is introduced as a bookkeeping device that keeps track of the accumulation of infinitesimal updates of a probability distribution. An instant is specified by a probability density $\rho(x,t)$, and the next instant is defined by

$\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$

The parameter $\Delta t$ is chosen so that the inferred fluctuations are homogeneous, and the asymmetry between prior and posterior gives the construction a built-in arrow (Caticha, 2010).

A related formulation on Gibbs statistical manifolds makes the same point geometrically. There the system’s macrostate is itself a probability distribution $p(x|A)$, the Fisher–Rao metric supplies the invariant notion of step size, and the identification

$T=\Delta t$

calibrates duration by the expected information distance traversed in one step. In that setting the slogan “the system is its own clock” is literal: there is no external timekeeper distinct from the evolving statistical state (Pessoa et al., 2020).

These definitions are therefore not interchangeable. Some identify entropic time with specific entropy or entropy production, some with an inferential update parameter, and some with an information-geometric duration. What unifies them is the replacement of uniform external time by a relational quantity derived from irreversible change, uncertainty, or state-space structure.

2. Thermodynamic clocks from irreversible entropy production

The paradigmatic thermodynamic example is free expansion of an ideal gas into vacuum. For $N$ identical, noninteracting particles initially confined to a region of radius $r_0$ in one dimension, the expansion law in conventional time is

$r=r_0+vt.$

An inside observer partitions the interval $d\tau = dS_{\mathrm{ir}}/N,$0 into $d\tau = dS_{\mathrm{ir}}/N,$1 cells of size $d\tau = dS_{\mathrm{ir}}/N,$2, so

$d\tau = dS_{\mathrm{ir}}/N,$3

The number of ways to distribute $d\tau = dS_{\mathrm{ir}}/N,$4 identical particles into $d\tau = dS_{\mathrm{ir}}/N,$5 cells is

$d\tau = dS_{\mathrm{ir}}/N,$6

and Stirling’s approximation gives

$d\tau = dS_{\mathrm{ir}}/N,$7

Defining the entropic time as the entropy per particle,

$d\tau = dS_{\mathrm{ir}}/N,$8

yields

$d\tau = dS_{\mathrm{ir}}/N,$9

With

$\rho(x,t)$0

this becomes

$\rho(x,t)$1

The resulting relation to conventional uniform time is nonlinear: at early times $\rho(x,t)$2 is approximately linear in $\rho(x,t)$3, while at long times

$\rho(x,t)$4

Accordingly, the entropic clock becomes logarithmic relative to the reference clock at late stages of the expansion (Martyushev et al., 2016).

The same authors later generalized the construction by taking the number of accessible microstates $\rho(x,t)$5 as primitive and postulating

$\rho(x,t)$6

Choosing $\rho(x,t)$7 again gives $\rho(x,t)$8. In their model with boson-type and fermion-type particles distributed over $\rho(x,t)$9 cells, the central implicit relation between $\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$0, system size $\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$1, and particle number $\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$2 leads to the condition

$\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$3

so monotonic increase of entropic time implies either growth in size at fixed $\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$4 or particle loss at fixed $\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$5. The same framework introduces an analogue kinetic energy and corresponding “entropic forces,” with inverse-square behavior in the superdense regime and constant behavior in the sufficiently low-density regime (Martyushev et al., 2018).

This thermodynamic line of work also explicitly notes analogous logarithmic links between intrinsic and external time in biology and cosmology. Martyushev et al. and Richardson and Rosen are cited for developmental time tied to entropy production in biological growth, while Milne’s kinematic relativity is cited for a logarithmic relation between “cosmic” and atomic times (Martyushev et al., 2016).

3. Entropic Dynamics and local temporal order in quantum theory

In Entropic Dynamics, time is introduced not through irreversible thermodynamics but through inference. The nonrelativistic construction begins with short uncertain steps in configuration space and a maximum-entropy assignment of the transition probability. Requiring spatially homogeneous fluctuations forces the Lagrange multiplier to take the form

$\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$6

so a displacement decomposes as

$\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$7

with drift

$\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$8

and fluctuations

$\rho(x',t+\Delta t)=\int dx\,P(x'|x)\,\rho(x,t).$9

Iterating these steps yields a Fokker–Planck equation, and the asymmetry between $\Delta t$0 and the Bayes-reconstructed reverse transition $\Delta t$1 establishes an inferential arrow of time. When the entropy field is allowed to back-react and an energy functional is conserved, the coupled equations for $\Delta t$2 and the phase reduce to the Schrödinger equation (Caticha, 2010).

Caticha’s separate treatment of the same theme emphasizes the same point in explicitly temporal terms. There, entropic time is “introduced purely as a bookkeeping device,” an instant is the density $\Delta t$3 itself, and the distinction between past and future arises because inference proceeds from prior to posterior. Standard unitary quantum dynamics then emerges from a framework in which the underlying time variable is epistemic rather than primitive (Caticha, 2010).

The field-theoretic extension replaces particle configurations by field configurations $\Delta t$4 in an infinite-dimensional configuration space $\Delta t$5 with metric

$\Delta t$6

Successive instants are distributions $\Delta t$7, and entropic time is calibrated so that field fluctuations remain homogeneous and stationary:

$\Delta t$8

In that sense, “the clock of entropic time is set by the field fluctuations themselves” (Caticha, 2012).

A further covariant generalization introduces a local notion of entropic time adapted to relativistic field theory. An instant is now a spacelike hypersurface $\Delta t$9 together with the epistemic state $p(x|A)$0, and duration is defined pointwise by normal deformations $p(x|A)$1. Compatibility with arbitrary foliations is enforced by Dirac–Kuchař–Teitelboim-style hypersurface-deformation brackets and the requirement of path independence. The result is a local-time Fokker–Planck equation and a local-time Schrödinger equation in which one clock is assigned to each spatial point rather than a single global parameter (Ipek et al., 2018). The curved-space formulation makes the same idea explicit: “time evolution consists of the accumulation of changes induced by local deformations” of spacelike surfaces, and the resulting dynamics is a non-dissipative diffusion constrained by foliation invariance (Ipek et al., 2018).

4. Cosmological and geometric formulations

One cosmological proposal distinguishes coordinate time $p(x|A)$2, assumed to flow at a fixed rate, from cosmological proper time $p(x|A)$3, whose flow is postulated to be proportional to the entropy of the causally connected region of the Universe. In practice the dominant early contribution is taken to be radiation entropy,

$p(x|A)$4

with $p(x|A)$5 in a Robertson–Walker model. The central hypothesis is

$p(x|A)$6

or, equivalently, a normalization of $p(x|A)$7 by its Big Bang value. For matter domination the model yields

$p(x|A)$8

When $p(x|A)$9 the flat case gives $T=\Delta t$0, but for spatially closed models the clock slows during expansion. Plotted against $T=\Delta t$1, the scale factor can then display positive curvature over the observational interval without any cosmological constant or dark energy. In this framework, the apparent acceleration is a kinematic effect of the varying clock rate, while the underlying geometry remains closed and finite (Clementine et al., 2017).

A distinct geometric program uses geometrothermodynamics to interpret entropy as a time parameter along thermodynamic geodesics. For an ideal gas in the entropic representation with $T=\Delta t$2, $T=\Delta t$3, the induced equilibrium metric can be written in logarithmic coordinates

$T=\Delta t$4

for which

$T=\Delta t$5

Geodesics satisfy

$T=\Delta t$6

so the ideal-gas entropy becomes

$T=\Delta t$7

that is, an affine function of the geodesic parameter. The second law then restricts admissible geodesics to the forward “adiabatic cone,” and the direction in which entropy increases provides the arrow of time. The same paper argues that this entropy-affine property is local for arbitrary equilibrium manifolds and for systems in the linear non-equilibrium regime under local equilibrium (Quevedo, 2024).

These cosmological and geometric constructions differ in mechanism. The former ties clock rate to horizon entropy within Friedmann dynamics, while the latter identifies entropy itself with an affine parameter on an equilibrium manifold. Both, however, replace a conventionally uniform temporal parameter by a quantity determined by thermodynamic state.

5. Information-theoretic and experimental formulations

A quantum-information version of entropic time is developed by replacing time with steadily read-out quantum correlations. In Miyake’s model, a hybrid quantum–classical array contains local qudits, memories, and a finite classical register $T=\Delta t$8 carrying past information. The key constraint is the data-processing inequality

$T=\Delta t$9

together with a “least conditional entropy” postulate enforcing

$N$0

at every elementary step. The clock is thus not an external parameter but a constant flow of accessible information encoded in correlations and limited by strong subadditivity. Unitary transfer operators arise precisely when this entropic bound is saturated (Miyake, 2011).

A related, but conceptually distinct, proposal defines entropic time as the ordering of irreversible information-acquisition events. In a bipartite decoherence setting, the reduced density matrix of a subsystem loses off-diagonal terms as environment states become orthogonal, and the subsystem entropy increases from zero to

$N$1

Entropic time is then the partial order of these information gains, in contrast with the reversible parametric time $N$2 of the Schrödinger equation. In this view, collapse is atemporal in the relativistic sense, whereas informational ordering supplies the experienced arrow (Vaughan, 2020).

The most direct experimental realization to date is an analogue Wheeler–DeWitt mini-universe implemented in a partitioned Bose–Einstein condensate. A barrier divides a well-isolated condensate into a bright sector and a dark sector, permitting entropy exchange through tunable coupling. The entropic time is defined along the bright-sector trajectory by

$N$3

with $N$4. Experimentally, $N$5 is strictly increasing except at turning points where $N$6 changes sign but no entropy flows, and the rate $N$7 is proportional to the entropy-exchange rate controlled by the barrier height. In the same system, re-expressing the Wheeler–DeWitt-like bright-sector equation in terms of $N$8 yields an entropic-time Schrödinger equation,

$N$9

which reduces to the usual local-time form when $r_0$0. In the high-barrier limit, entropy flow vanishes and $r_0$1 saturates, described in the paper as a thermodynamic “heat death” (Barontini, 9 Sep 2025).

6. Debates, adjacent uses, and later extensions

A major contemporary controversy concerns whether entropy can define time itself or only its arrow. Finberg argues that identifying time with entropy is a category error: the fundamental equations of mechanics, quantum theory, and general relativity presuppose time as the parameter of change and are invariant under $r_0$2, whereas entropy increase requires an already ordered pair of times $r_0$3. On this view, entropy provides a statistical bias along a pre-existing temporal axis but does not create that axis. Even in a heat-death universe, temporal structure persists through quantum-vacuum correlators such as

$r_0$4

and through the Lorentzian metric

$r_0$5

which continue to order timelike separations without supplying a preferred orientation (Finberg, 19 Aug 2025).

A distinct adjacent literature uses entropy not to define a clock variable directly but to quantify uncertainty about time. For a quantum system with Hamiltonian $r_0$6, one considers a classical time register $r_0$7 or a displacement label $r_0$8 attached to unitary evolutions $r_0$9 or $r=r_0+vt.$0. The resulting entropic energy–time relations take forms such as

$r=r_0+vt.$1

and, in a more general algebraic setting,

$r=r_0+vt.$2

These works operationalize time uncertainty through conditional entropy of a classical register rather than by identifying entropy with time itself (Coles et al., 2018, Bertoni et al., 2020).

The terminology has also migrated into domain-specific applications. In generative diffusion modeling, entropic time is defined by the conditional entropy

$r=r_0+vt.$3

and used as a time reparameterization so that each sampling point contributes an equal amount of information. The same work introduces a rescaled entropic time

$r=r_0+vt.$4

shows invariance under the original time scale, and reports improved inference performance for pretrained EDM2 models, especially in the few-NFE regime (Stancevic et al., 18 Apr 2025). In psychophysics, a 2026 proposal defines an internal entropic time by

$r=r_0+vt.$5

derives a local time metric scaling as $r=r_0+vt.$6, and couples it to deformed leaky integrate-and-fire dynamics to model subjective time dilation and compression (Weberszpil et al., 28 Jun 2026).

Taken together, these debates and extensions show that “entropic time” is best understood as a research program rather than a single doctrine. In some papers it is a thermodynamic clock based on entropy production; in others an inferential ordering parameter, a geometric affine parameter, a relational variable extracted from entropy exchange, or an entropy-based uncertainty measure. The strongest point of convergence is not an agreed ontology of time, but the repeated claim that temporal structure can be reconstructed from irreversible change, informational asymmetry, or state-space geometry rather than presupposed as uniformly given.