The arrow of time has a constructive role influencing entropy production.  In biological systems, in has been recently demonstrated that molecular stochasticity is central to understanding phenotypic heterogeneity and the stability of cellular proliferation (Kiviet).  Such a concept pertains also to the brain, because synaptic transmission involves molecular processes that display stochastic (random) properties governed by Brownian motion (Ribrault; Wang).  Non-equilibrium steady-state systems are regulated by two kinds of directional entropies: the (forward) standard entropy per unit time and the (backward) time-reversed entropy (Gaspard).  The difference between the latter and the former quantifies the entropy production in the system (the brain, in our case), according to the formula:

in which stands for the entropy production of non-equilibrium steady state, hR (Р) for the time-reversed entropy and h (P) for the τ-entropy (i.e., the standard entropy per unit time τ).  P is the coarse graining or partition. 

The formula says that the forward entropy exhibits less randomness than the backward one, so that the entropy production is higher than the one produced by the sole standard entropy per unit time (see the dotted line in Figure).  This phenomenon is due to non-equilibrium steady-state constrains, together with the randomness generated by the stochastic process during system's time evolution: they impose stochastic boundary conditions which break the time-reversal symmetry, leading to the unexpected amount of entropy production over the course of time.  It must be emphasized that the entropy production of non-equilibrium steady state is indirectly related to the thermodynamic entropy via the Boltzmann’s formula and to Shannon’s and Kolmogorov- Sinai’s ε-entropy via the partition P of the phase space of the system into cells ω of size ε (Cencini).  The time-reversal asymmetry has thus two opposite effects: on the one hand, the passage of time introduces biases in the probability and influences the evolution of the system capable of memory; on the other, the non-equilibrium system goes towards a progressive increase of -both Gibbs and Shannon entropy and, consequently, a lower rate of information production, meaning that the brain is ageing and history-dependent (Sherrington).

In search of possible psychological correlates of time-reversal asymmetry, we notice that also in one of the most successful models of brain function, the free energy principle (Friston), the time has in important role.  In Friston’s framework, entropy is simply that of the agent’s own recognition density.  The time is incorporated in the recognition density: indeed, the latter is the probabilistic representation q(δ|μ) of the causes of sensory input, in which the internal states of the brain are μ(t) and the causes δ⊃ {x˜,θ, γ}, comprising hidden states x˜(t), parameters θ and precisions γ controlling the amplitude of the random fluctuations.  As times goes on, the recognition density\entropy decreases due to the time-reversal asymmetry, leading to habituation, fainting of memories, vanishing of thoughts.  But the cognitive functions need a lower level of both thermodynamical and information entropies: indeed the microdiversity converting resilience to complex adaptive systems must be given by a traveling holon staying in a very unstable and low entropic state on the global maximum of the attractor landscape (Chu; Carlson).

In conclusion, the non-equilibrium steady-state brain shows an unexpected increase of both thermodynamic and information entropy, due to the time-reversal asymmetry linked with backward time entropy.  It may lead to the fainting of memories observed in humans with time passing. 



Schematic representation of Shannon’s entropy and entropy production, Modified from Shannon (1948) and Gaspard (2005).  The Shannon’s entropy is plotted as a function of the random variable p, in the case of two possibilities with probabilities p and (1-p). 

The dotted line shows the entropy production (in a non-equilibrium steady-state) versus the mean number of particles A, according to Gaspard (2005).  The particles A are associated with the left-hand side reservoir for a diffusion process of Brownian particles in a pipe, while the right-hand side reservoir has its concentration fixed at B = 100 (see Gaspard for further details). The thermodynamic equilibrium happens at A = B = 100, where the entropy production vanishes.In our simplification, the line A is truncated at the number of particles A=200. 




1.       Carlson, J. M. & Doyle, J. (2002). Complexity and robustness. Proceedings of the National Academy of Sciences, 99, suppl. 1:2538-45.

2.       Cencini M, Falconi M, Olbrich E, Kantz H and Vulpiani A 2000 Phys. Rev. E 62 427

3.       Chu, D., Strand, R. &Fjelland, R. (2003). Theories of Complexity: Common Denominators of Complex Systems. Complexity, 8(3), 19-30.

4.       Friston K (2010) The free-energy principle: a unified brain theory? Nat Rev Neurosci 11(2):127-138. doi: 10.1038/nrn2787 Gaspard P.  Brownian motion, dynamical randomness and irreversibility.  2005 New J. Phys. 7 77 doi:10.1088/1367-2630/7/1/077

5.       Gaspard P.  Brownian motion, dynamical randomness and irreversibility.  2005 New J. Phys. 7 77.  doi:10.1088/1367-2630/7/1/077

6.       Kiviet DJ, Nghe P, Walker N, Boulineau S, Sunderlikova V, Tans SJ.  Stochasticity of metabolism and growth at the single-cell level.  Nature 514, 376–379 (16 October 2014) doi:10.1038/nature13582

7.       Ribrault,C.,Sekimoto,K.,Triller,A.(2011).Fromthestochasticityofmolecularprocessestothevariabilityofsynaptictransmission. Nat.Rev.Neurosci. 12, 375–387. doi:10.1038/nrn3025

8.       Shannon CE (1948) A Mathematical Theory of information.  The Bell System Technical Journal, 27:379-423, 623-656 

9.       Sherrington,D.(2010).Physicsandcomplexity. Phil.Trans.R.Soc.A 368, 1175–1189. doi:10.1098/rsta.2009.0208

10.    Wang H, Wang B, Normoyle KP, Jackson K, Spitler K, et al.  Brain temperature and its fundamental properties: a review for clinical neuroscientists.  Front. Neurosci., 08 October 2014 | doi: 10.3389/fnins.2014.00307