Irreversibility

Irreversibility is a fact of life. A hot cup of coffee sitting on the counter of your kitchen gets cold. The opposite never happens spontaneously. A gas released in vacuum quickly expands to fill the whole volume. But this simple fact of irreversiblity is actually a conceptually messy thing in physics. The following is my attempt to explain this rather obtuse concept in a non-mathematical and jargon free form.

In order to motivate what I want to say in this post, I ask you to try and do the following Gedanken experiment (German for thought experiment) in your heads. Imagine that I have turned off gravity for the moment. Imagine a big box in which I put two balls with some kinetic energy. What would they do? They would move with constant velocity in the direction of their velocity. Further I assume that if they hit one of the walls of the box or each other, they will rebound elastically (i.e., will not loose any of their energy). So, if I watch these balls for a while, they are just going to rattle around for ever. Now, I make a movie of this experiment and then show it to somebody else. But, I play the movie backwards in the rewind mode. Will they be able to tell that I am playing the movie backwards? The answer is no.

The system we used in the above experiment can be thought as a minimal model for a gas in a container, albeit a gas with only two molecules. Now, note two things about the above model system. The total amount of energy in the system is the same at all times, this is called “conservation of energy”. Next, note the claim that I cannot distinguish between the movie of the above experiment played forward in time and backward in time. This is called “time reversal invariance”. These two properties are fundamental properties of real physical interactions between molecules. All of microscopic physics has time reversal invariance [1].

Now, let us do another experiment. Again, I have turned off gravity. I take a big room, I seal it off and then vacuum it, in that I remove all the air from the room. In the middle of this room I have a little nozzle that when I start this experiment, releases a small amount of a pink gas. Now, if I wait a while, what will happen? The pink gas will expand and fill the whole room. And hence if I make a movie of this experiment and show it to somebody in the rewind mode, they will be able to tell immediately that I am showing them the movie backwards for a gas that fills the whole room will appear to spontaneously go back to a small volume in the middle of the room. And everybody knows that that cannot happen. Yes?

Next, let us reformulate the second experiment in terms of the model for a gas we used in the first experiment. So what am I doing? I am releasing a large number (about 10000 billion say [2]) of pink balls, each with some kinetic energy into a big box and watching them for a while. The microscopic physics for this system is the same as in the first experiment, i.e., it has “time reversal invariance”. But, when I look at a system with a large number of balls, I “know” the direction of time. So, what is wrong? What did I miss?

The first thing you might ask is “Did you do the calculation for the 10000 billion balls to see that the theory predicts that the system still has time reversal invariance?” The answer to this question is that nobody and no computer can do this calculation. Then you say “Ah! Then you don’t know what your theory says so the whole point is moot!” Well, you would be right but for the fact that the mathematician Poincare proved that if you take a finite system (i.e., a system in a box say) with finite energy, then it will always come back to where it started from [3]. In the case of our experiment, Poincare’s theorem says that the gas that fills the room will eventually all come back and sit in the middle of the room again. But we know this does not happen. So, ask again, what is wrong? The catch here is that you have to ask how long does it take for the system to come back to where it started from. The answer to this question, for the case of the billions and billions of balls that is our gas is a time greater than the age of the universe! So, no matter how long you watch it, it is never going to come and sit in the middle of the room because you can never watch long enough! So, the message here is that the irreversibility that we ubiquitously observe in the world around us is an accident of the time scales over which we conduct the said observations.

Ok, we explained the paradox. But so what? Physics is supposed to be a mathematical model for observed phenomena. And the observed phenomenon here is that the spontaneous expansion of a gas is irreversible on the time scales that are relevant to life on earth. Poincare Recurrence might be a fact of life, but is irrelevant for anything we observe. So, what physics should give me is a rule of thumb that some things will happen (eg., a gas released in vacuum will expand to fill all available space, or a hot cup of coffee placed in a cold room will get cooler) and some things won’t (eg., gas in a room will not all spontaneously go and sit in one corner of the room, or coffee will not get hotter by taking energy from the cold room and hence making it even colder). This rule of thumb is called the Second Law of Thermodynamics that we all learnt in high school.

In the form we learnt in high school, this law is usually stated as “Heat always flows from a body at a higher temperature to a body at a lower temperature”. But a more general statement of this law would be that any system evolves so as to maximize its randomness. It is easy to see that this law explains why a gas will expand to fill the whole room. A given number of molecules occupying a small volume is less random than the same number of molecules occupying a larger volume right? It is rather more obscure to see how the two forms of the law (the first one more naturally explains the coffee scenario while the latter more naturally explains our Gedanken experiments) are equivalent, but it can be shown that they indeed are. But the point to note is that as I said earlier, this law, as is all of thermodynamics, is a rule of thumb to explain observed reality on length scales and time scales relevant to us. Deriving this from fundamental theory is a major mathematical headache addressed by all manners of scientists, mathematicians studying dynamical systems, physicists studying statistical mechanics and so on, with various degrees of success [4]. But clearly, it works and works very well! So we use it anyway.

(For my few regular readers, don’t worry, this post is an anomaly, not the new norm!)

Caveats for experts

[1] CP violating weak forces are not in my picture, I am living in a QED+classical gravity world of macroscopic physicists.

[2] Actually, an Avogadro number of molecules.

[3] I know that Poincare Recurrence theorem talks about approach within an arbitrarily small neighborhood of the initial condition and is stated in terms of bounded orbits, but I did not know how else to say this in plain English.

[4] The basis of thermodynamics in Statistical Mechanics is well established and there is a well developed theory of Hamiltonian dynamics that lets you ask such things as ergodicity and mixing in the trajectory space to substantiate domain of validity of the postulates of Stat Mech, but all this is rather esoteric stuff is n’t it? Or can we state it in plain English in an accessible way?