To say an ‘Arrow of Time’ is to say that time itself is linear and one-directional. In thermodynamics, the second law dictates that everything moves in one-direction from a state of order towards disorder and entropy is the statistical measurement of this asymmetry in an isolated system. The universe is, for instance, this isolated system and as a consequence it is impossible to reverse this arrow of time and travel backwards just as much as the continuity of disorder will never decrease. Newton’ equations and other laws in physics, however, can be reversed and thus this ‘order’ is arrived from a state of equilibrium as it moves forward toward disorder.
We simply understand the past through entropy and yet, while this may be statistically correct, the Poincare’ recurrence theorem in open systems prove that thermodynamic system may actually be paradoxical. The theorem purports that after a length of time the system may return back to its original or close to its original state, such a hot cup of tea eventually reaching room temperature. Statistical physics that attempts to measure the universe as a finite and objective physical entity together with the evolutionary patterns of nature, reached an impasse since the cosmological theory of the universe with the Big Bang started with chaos and eventually formed this smooth, ordered state that works in contradiction to the second law leading to an reductio ad absurdum. Unless we assume that the big bang itself occurred as a ‘fluctuation’ – indeed, a very rare and unique one – and is thus an addition contained inside one meta-system where a number of universes exist.
Ludwig Boltzmann did not believe in the reversibility of statistical thermodynamics; I become conscious or self-aware and thus experience a non-equilibrium with nature or the external world because of a random fluctuation in my brain, but intelligence is only available in my brain and so why everything else in this system – body, organs – and thus consciousness is a random fluctuation. If I want to bake a cupcake, the ingredients – such as flour – can be reversed back to wheat, soil, planet earth, milky way, universe etc &c., – and rather than saying that I need a universe to create a cupcake or that I need to create a universe to make a cupcake, it is easier to explain that a cupcake is made from a random fluctuation. Ultimately, however, we do need a universe, milky way, planet earth, soil, wheat, and flour to make a cupcake and thus every system requires time to move forward as well as backward.
Boltzmann’s entropy formula S = k log W describes the statistical domain of thermodynamic systems [log itself aids with minimising the size of the universe with W being the number of microstates giving a probability at macro-level] and that entropy is conserved by the monotonic function of ordered sets as the microstates increase. Any corresponding change at macro-level is causally connected to a change with a microstate within the system, but this interconnected dependence between the two states naturally shows that the macro-state itself contains maximal entropy.
That is, thermodynamic equilibrium of system is consistent with the constraints of the second law of thermodynamics and that the formula is the statistical evidence of this. In its simplest, ergodicity is merely analogous to ascertaining the averages of behaviour within a system, measuring transformations, recurrence, arbitrary convergence etc &c. or quite simply the dynamics and that over time the probability of visiting every required state occurs. A macro-state in equilibrium is largest in size, thus over ‘time’ [that is, time-average probability] the system spends visiting the phases as it reaches this equilibrium and thus maximum entropy is, well, ergodic. Even so, it is still incredibly difficult resolving arbitrary estimates coupled with the fact that cases involving the second law of thermodynamics do not necessarily require erodicity at all. But when considering the constituency of time in this framework, the idea that the direction or arrow of time will eventually lead a system toward maximum entropy and ergodicity may, in reverse, explain time.
Ergodicity itself is somewhat Epicurean, not to say that it has any connection with Epicurus’ Nature of the Cosmos, but rather the philosopher himself – more notably adhered by Lucretius – believed that it is a mark of an intelligent mind to think of multiple possibilities – from the absurd to the rational – so as to identify and explain a solution to a cosmological problem; even occasionally, multiple descriptions can prove a theorem adequate and inadequate at the same time. Lucretius’ cosmological phenomenology is based on his thought experiment regarding infinite space, whereby should one travel to the end of the universe and throw a spear through it, what would happen to the spear? Either it will hit it and fall, or it will go through the boundary – that the boundary of a finite universe is ultimately illusory – and toward another space that we are not aware of; what this would mean is that all possibilities and possible worlds outside of the finite space that we understand is actually possible.
Therefore the universe is infinite and according to Newton this must be true; his failure, however, was the proposition that the universe was static purported by his assumption of the stars being fixed relative the inertial frame, namely because the distribution of mass would be unstable. The problem of symmetry, however, regarding a state where the universe is accelerating, is how the direction for which this acceleration is determined. In order to substantiate the validity that there is actually a physical system, the universe requires isotropy and since we can acknowledge that when we look out to the universe that in every direction we can observe the CMB radiation, one can conclude that it may very well be a symmetric space.
Nevertheless, for the sake of avoiding the likelihood of falling down the existential rabbit hole before becoming overwhelmed by the vanity of, well, everything, let us assume that the universe can be modelled as a dynamical system, contained in an isotropic, homogenous and maximally symmetric but statistically within a finite structure and governed by an arrow of time, and in doing so the analysis of erodicity and entropy within such a model seem almost possible.
The problem is that, if the arrow of time purports that time itself is moving forward in one direction, that the universe is expanding alongside time as it reaches its maximum state of high entropy, it would mean that therefore the universe had a past and so to not defy the second law of thermodynamics, the early universe would have to be at a state of low entropy. In an environment where the observable universe is much denser or smaller in the past – since the universe is expanding – it would logically imply that it was hotter and the pull stronger. How is it that in that macroscopic parameter consisting of a hot and dense environment instead was smooth and cool?
This does not make sense since the early universe was in a state of equilibrium which, given the calculations above, must uphold the thermal law of being in a state of high entropy. In addition, temporal asymmetry works in contradiction to the second law of thermodynamics; motion cannot function without time, it would be like matter frozen in a dimensionless space or swallowed in a blink beyond the event horizon. Time’ arrow works in a manner that directs motion forward, evolutionary of sorts and adapts to the processes within its environment in an attempt to find a state of equilibrium.
In classic thermodynamics, the joule [free] expansion – where within an adiabatic container enclosed with monatomic gas molecules and no energy or thermal properties – the gas densely kept to one side of the container with a closed partition between another empty container that has been vacuumed of any properties at all and therefore completely empty, when the partition is opened and consequently the gas in one container increases in volume and expands into the other, the pressure of the gas that had been densely kept in the other compartment diminishes [like blowing up a balloon with helium gas and then letting it go; the gas is released from the balloon with the rubber shell flying about the place in an awkwardly loud and flatulent manner].
There is no pressure or work, ΔU = q + w = 0 but nevertheless there were changes [in consideration of ideal gas] in temperature and therefore PV=nRT whereby the pressure and volume equates to a constant of the gas and the temperature, so the first law regarding the conservation of energy in thermodynamics remains valid. The ergodic hypothesis by Boltzmann was formulated to prove in principle the determination of the distribution of gas molecules and their kinetic speeds in his equipartition theorem, which is mathematically ascertaining the energy of any given physical system through the distribution of generalised coordinates and momenta.
The second law of thermodynamics contains the interesting problem vis-à-vis this very blog post, that the law governs the exchange of thermal contact and gradual arrangement toward a fixed equilibrium; that is, the natural evolution of any given system is determined to a state of equilibrium. Once the partition is open and the gases are dispersed, they spontaneously find a state of equilibrium and do not randomly paste themselves to the ceiling of the container etc &c. How can a hot cup of tea become lukewarm as it cools to room temperature and thus asymmetric as it reaches a state of equilibrium with its environment? Or is that a deductive fallacy?
Quantum entanglement is an interesting method of understanding the arrow of time in this context. The uncertainty principle in quantum mechanics asserts that measuring the position of a particle and its momentum is never accurate, furthered in confusion with evidence that sometimes interaction between two particles merge or entangle to form a ‘oneness’ that dictates the momentum and position one to the other that they are no longer two separate particles – though physically it is so –nevertheless communicating invisibly one to the other as a combined force. It is of interest to me where the interaction prior to the amalgam between the particles peaked at a derivative equal to zero, namely the very point where particles enjoin to become a state where they can no longer be classified as autonomous. Reaching this balance of connectivity between particles from a pure particle state to a combined oneness in perfect equilibrium as it relaxes into its new and unchanging form is the real parameter that works comparatively to the notion of thermal equilibrium and thus the evaluation of thermodynamic properties.
It appears that no matter where I am in the universe, I will still get the same answers to the same equations and the physical world would appear to me, well, to be the same in every direction. That is, the symmetry of the expansion rate is homogenous confirmed to a degree through Hubble’s Law, which is the velocity between two galaxies being equal to the Hubble parameter times the distance V=Hod and verifies that objects would appear to be expanding outward relative to the observer; the measurement of the radial velocity determined by the redshift. Thus galaxies are moving away and galaxies even further still at a much faster rate explained by the fact that should both the source and the observer be stationary, there would be no time differentiation or delays viz., the time for the wavelength to reach the observer, hence the Doppler effect.
When thinking about the cosmological redshift, whereby light that has been emitted from a distant galaxy reaches us on earth, calculations of the spectral features of photons namely λ =h/p requires attentiveness on how the light itself will shift from the frequency it had when emitted to the frequency we measure when receiving it, that is, the momentum and time it takes for the wavelength to reach the observer, the evolution of this process changes as the photons are stretched. Physicists have thus determined that the universe is not only expanding but also accelerating.
It would seem that the universe is expanding whilst galaxies themselves remain static. When photons emitted from a distant galaxy reach us the observer, the distance and velocity of the wavelength with the time it takes to us from the source is quantified by the Hubble constant times the distance between galaxies. The asymptotic nature of ∆t would purport that the atomic properties in space and thermal energy interact with time in a manner that will continuously interfere in the process of reaching absolute zero, and as stated previously, even a vacuum state still contains energy even though extremely low. Measuring time during the inflationary epoch remains questionable, even with the capacity to measure the smallest possible unit of time through Planck [5.39 × 10^-44 s] whereby probabilities are the only reality attributed to the questionable state of time. Perhaps the total entropy in the universe is already infinite, in which case it was always infinite.
How we experience time remains an unexplained phenomenon. If, indeed, time must move both forward and backward, perhaps the equilibrium that I experience in ‘now’ is really both past and future working in perfect uniformity rather than some random existential fluctuation. Perhaps my past can speak to my future and that is the mystical experience of prophesy? That there is no ‘beginning’ or ‘end’ except this singularity itself, namely God who is ‘alpha and omega’? Or maybe the universe is simply a brain!
 Don S. Lemons, A Student’s Guide to Entropy, Cambridge University Press (2013) 72
 See Lucretius’ cosmology and use of the Principle of Plentitude briefly explained in Michael J. White, Agency and Integrality: Philosophical Themes in the Ancient Discussions of Determinism and Responsibility, Springer Science & Business Media (2012) 4
 Philip de May, Lucretius: Poet and Epicurean, Cambridge University Press (2009) 27
 Clement John Adkins Equilibrium Thermodynamics, Cambridge University Press (1983) 162
 Peter Atkins, Julio de Paula, Ronald Friedman, Physical Chemistry: Quanta, Matter, and Change, OUP Oxford (2013) 576
 David Darling, The Universal Book of Mathematics: From Abracadabra to Zeno’s Paradoxes, John Wiley & Sons (2004) 139. See Robert Heinlein’ All You Zombies
 K.V.S.Gnaneswara Rao, Engineering Physics, S. Chand Publishing (2008) 38