# In thermodynamics, time matters or not matters? Actually, this is a paradox

## Beginning

The 2nd law of thermodynamics states that time matters, because things become more disorderly with time; but the concept of time symmetry states that time doesn’t matter, as anything that happens by some physical process can be reversed by the opposite physical process.

## Before Loschmidt

In 1874, two years before the Loschmidt newspaper, William Thomson defended the second law against the statute of limitation.

## Paradox

**Loschmidt’s paradox**, also known as the **reversibility paradox, irreversibility paradox or Umkehreinwand** is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This contradicts the (almost) reverse symmetry of all the low-level basic physical processes that are known over time, as well as the effort to leave the second thermodynamic law that defines the behavior of macro-thermal systems. Both are principles recognized in physics, with solid observational and theoretical support, yet appear to be in conflict; so the paradox.

## Origin

Josef Loschmidt’s criticism was provoked by Boltzmann’s H-theorem; this used kinetic theory to explain the increase in entropy in an ideal gas in a non-equilibrium state when the gas molecules were allowed to collide.

In 1876, Loschmidt pointed out that if there is a motion of a system from time t0 to time t1 to time t2 that leads to a steady decrease of H (increase of entropy) with time, then there is another allowed state of motion of the system at t1, found by reversing all the velocities, in which H must increase. This revealed that Boltzmann’s basic assumptions, molecular chaos, or Stosszahlansatz, all particle velocities were completely unrelated, didn’t follow the Newtonian dynamics. One can assert that possible correlations are uninteresting, and therefore decide to ignore them; but if one does so, one has changed the conceptual system, injecting an element of time-asymmetry by that very action.

The laws of reversible action cannot explain why we have experienced our world so far in the case of a relatively low entropy (compared to the equilibrium entropy of universal heat death); and in the past there was even less entropy.

## Arrow of time

Any process that happens regularly in the forward direction of time but rarely or never in the opposite direction, such as entropy increasing in an isolated system, defines what physicists call an arrow of time in nature. This term only refers to an asymmetry observation over time; This does not mean suggesting an explanation for such asymmetries.

The paradox of Loschmidt is equivalent to the question of how time-symmetrical fundamental laws can be a thermodynamic arrow of the given time, because time-symmetry implies a reverse version that is similar to that of any other process compatible with these fundamental laws.

## Dynamical systems

Current research in dynamic systems provides a possible mechanism for obtaining irreversibility from reversible systems. The central argument is based on the claim that the correct way to study the dynamics of macroscopic systems is to study the transfer operator corresponding to the microscopic equations of motion. It is then argued that the transfer operator is not unitary (i.e. is not reversible) but has eigenvalues whose magnitude is strictly less than one; these eigenvalues corresponding to decaying physical states. This approach is fraught with various difficulties; it works well for only a handful of exactly solvable models.

Abstract mathematical tools used in the study of energy-consuming systems include definitions of mixing, immigrant clusters and ergodic theory in general.

## The Big Bang

Another way to deal with the paradox of Loschmid is to see the second law as an expression of a set of boundary conditions in which the time coordinate of the universe has a low entropy starting point: the Big Bang. From this perspective, the arrows of time are completely determined by the direction away from the Big Bang, and the maximum entropy of a hypothetical universe with the Big Bang has no time arrows. Cosmic inflation theory tries to explain why the early universe has such a low entropy.

## Comments