Physics about equilibrium is a accomplished grand building now. Even isolated systems evolve toward equilibrium state, which is usually boring and irrelevant to the living world around us. From the molecular transportation through cell membranes, molecular motor walking on the microtubules,  to ear cell’s active resonance with the incoming sound wave, they all involve active process accompanied by energy consuming. The key issue in biophysics is the study the system which changes properties with time. i.e. a system far away from equilibrium. We want to know what the relationship is among entropy, energy and time arrow. How does highly organized structure of life come into being while the second law of thermodynamics predicts that our universe should be totally in chaos?

Laplace demon and time arrow in the deterministic system

After an apple luckily hit great Newton’s head and a complete mechanics system was set up, people hailed that a new era of enlightment was coming. With the Gravity Law and the initial position of the earth and the moon, we can predict where the moon will be. So Laplace proposed that if his smart demon knew the location and momentum of every atom in the universe then it could use Newton’s laws to know the history and future of the whole universe. Believing in causality and deterministic system was the virtue of every physicist in 19^th century. Even Max Planck, who initiated the discovery of Quantum Mechanics, still believed this concept. However, with the discovery of the chaos in the dynamic system, people realize that Laplace demon will soon know nothing about the universe even it knows the law and the initial conditions.

If a rock is moving in a conservative force field, we can NOT tell it is moving from now to future, or verse visa. Because the second law of Newton is about the second time derivative, we put F(x, -t) instead of F(x, t) and get exactly the same answer. So we can not define a time arrow in the deterministic system.

 

Isolated complex system in Non-equilibrium and time arrow

However, when we move to a system with large number of same particles, even each particle still follow a deterministic law, it will interact with so many other particles and soon it will lose its memory of the initial condition. So Laplace demon won’t help us to understand this system. But because it is a large system with the same particles, there is some structure of the randomness in this system. We all know that from the second law of thermodynamics, particles will occupy as much space as possible. If we look at the system at two different time, we know that the state with lower entropy happens before the state with the higher entropy. So in a complex system, we can decide the time arrow from the entropy of the system.

The current development in physics even allow us to do more about deciding time arrow.[1]. Supposing that we have two tube with the same ideal gas, one is pushing from left to right, another is pull from right to left. If we don’t look at the arrow above the graph, can know which is which.

At the first thought you may think it is impossible since the state of two system are exactly the same. However, if we assume that they are in Non-equilibrium state, then we surely can tell by its density distribution.  The higher density near the left wall is the case pushing to the right, the higher density near the right wall is the case pulling to the left.
 

Using this logic, we can do a think experiment: we can put an infinite thin wall in a container with ideal gas,

I) put it in the middle and push it quickly to the 1/3 to the right;  

II) put it the 1/3 to the right and push it quickly to in the middle;  

Can we distinguish the final state of case I and the initial state of case II? Yes, we can. Because the probability of ideal gas density on the both side of wall is ½ and ½; but in the second case it is 2/3 and 1/3. Based on this idea, we can derived a beautiful equation between Irreversibility and Energy dissipation.

 

D here is  Kullback-Leibler entropy.

The old second law of thermal dynamic is only energy dissipation is bigger than 0. This equation gives a more accurate lower bound for the second law. It also clearly shows that:

Irreversibility = Energy dissipation

Based  on this,  Juan M.R. Larrondo also have an article to criticize Feynman's analysis of the ratchet as an engine. More interesting topics about this equation will be introduced in the next post.

Reference:

[1] R.Kanal, PRL, 98, 2007

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