The Entropy of a Markov Chain

Clausius (1865)¹ defines a amount called entropy. By decomposing beingness processes arsenic a concatenation of engines, he shows that entropy ever increases for irreversible processes. For reversible processes for illustration Carnot's perfect engine, the alteration successful entropy is zero. But erstwhile an irreversible process occurs, entropy tin ne'er alteration unless power is applied to a system. This is what is known arsenic the 2nd rule of thermodynamics.

And whilst entropy itself whitethorn not beryllium measurable pinch a thermometer aliases ruler, it is still a useful conception since we tin cipher derived quantities from it that are straight measurable.

I've written antecedently rather vaguely astir 'life arsenic entropy'. This was an thought motivated by Schrödinger (1944)² done the conception of negentropy. Through negentropy, life seems to support bid by feeding connected the power astir it, and reducing its section disorder. But up until now, I've been confused astir what this really intends successful detail.

And truthful 1 measurement I'm trying to understand this is done models. One attack mightiness beryllium to make artifact models and past fig retired a mathematically accordant measurement to specify entropy successful those systems. Then, you could simulate the exemplary and person a amended thought of really the entropy evolves.

One specified artifact exemplary that is useful to get an intuition is Dyson's artifact exemplary of a cell. Dyson's artifact exemplary of a compartment is simply a Markov concatenation that settles into 1 of 3 equilibrium states, 2 of which are the 'life' and 'death' states. But if we attraction astir the vague conception of 'life arsenic entropy', past we should beryllium capable to make a meaning of entropy to use to the Markov chain.

But the conception of entropy arsenic primitively defined by Clausius was a usability of activity and temperature

And truthful it is unclear really to subordinate the 2 ideas together. Is location a different measurement to specify entropy successful the discourse of Markov chains that gives you the aforesaid result, accordant pinch beingness ideas?

Well, 1 measurement to onslaught this is by looking astatine the activity of Boltzmann, who quantified the narration betwixt entropy and the number of imaginable states successful a system.

Suppose you return a strategy and past observe a system's authorities variables, for illustration temperature, unit and volume, utilizing various instruments. Presumably, location are a number of different configurations of the beingness strategy that are compatible pinch it being successful that state. And if entropy is related to the conception of bid and disorder, past presumably entropy would beryllium higher if location were much imaginable states.

Here is simply a vague example. If you person a state of identical molecules astatine absolute zero, fixed successful position, pinch each astatine zero velocity astatine zero temperature, you person less possibilities for what the strategy could beryllium for the state to beryllium successful that state. Whereas if we had a heated state astatine a basking temperature, location are galore configurations that a authorities mightiness beryllium successful arsenic a result.

Temperature is higher, entropy is higher. Is it immoderate coincidence that the number of imaginable states increases too?

It turns retired location is simply a measurement to make these ideas rigorous. Boltzmann showed that entropy tin beryllium written arsenic a usability of the number of states that a strategy tin beryllium in, if we fixed what the macro authorities variables were. That intends if we could count each the states that were imaginable successful a strategy astatine fixed temperature, unit and volume, past we could cipher the entropy.

This is written successful the legendary equation below, which is besides amazingly elemental — that the entropy of a strategy is the logarithm of the number of states, multiplied by a fixed changeless called the Boltzmann constant.

Whilst the consequence is elemental it's wholly non-obvious. Boltzmann creates a exemplary of state wherever each particle only tin beryllium successful a group of finite velocities and positions and past tries to return the limit. The impervious is not easy and mathematicians are still figuring retired really to decently rigourise the activity of Boltzmann.

But anyway!

To make this illustration moreover much concrete, let's effort to compute the entropy of a basal system, fixed we cognize what the power (macrostate) is. Consider Curie's exemplary of a magnet. In this model, our beingness abstraction consists of a finite number of electrons, which tin either beryllium rotation up aliases rotation down. In models of magnets, power (or work) is done erstwhile the spins are misaligned pinch the guidance of the magnetic section that the metallic is placed in.

And truthful this suggests that if we had a postulation of atoms, the power is simply a usability of really galore atoms are aligned pinch rotation up vs rotation down.

Ok, truthful suppose we had a very, very mini strategy of 5 atoms. And suppose that we measured the power arsenic E = 1. First, we request to fig retired really galore atoms request to beryllium rotation up and rotation down to person this configuration. The evident reply is 3 rotation up and 2 rotation down because

We could besides trivially lick this utilizing the constraint that

n↑+n↓=5, so

And the configurations we could person are the following. Each configuration has 3 up spins and 2 down spins, but you tin put this successful 10 ways. The sketch beneath lists each of the possibilities.

This is 5 take 3 combinations, which is 10.

Those 10 configurations are the full contented of the macrostate "E = 1".

This intends that Boltzmann would explanation the entropy of the strategy as

or log₂ 10 = 3.32 bits. And truthful successful this lawsuit we tin spot the entropy arsenic a usability of energy. If we crippled retired entropy arsenic a usability of the energies we get this graph:

So now we've learnt really to compute entropy from the full number of microstates that a strategy tin have. Ok, truthful really do we count the entropy of a Markov concatenation process for illustration successful Dyson's compartment model?

In Dyson’s artifact exemplary that we person N sites, and each tract could beryllium successful 1 of 3 states — empty, active, aliases inactive. Given immoderate rules, successful my past station I showed that this converges to an equilibrium. The equilibrium values are a analyzable usability of fixed points, but they do exist, and the probability of being empty, inactive aliases progressive does converge to a fixed value.

This equilibrium constituent allows america to specify the entropy of a Markov concatenation done a counting argument. Say we had an equilibrium lawsuit wherever we had a 1/2 chance that a authorities is empty, a 1/4 chance that a authorities is active, and a 1/4 chance that a authorities is inactive.

This intends if we had 8 sites, past we would person 4 that were empty, 2 that were active, and 2 that were inactive. So to measurement the entropy, we now request to count each of the configurations of this happening. This is shown below, pinch each of the arrangements.

Counting them is the multinomial coefficient — take which 4 of the 8 sites are empty, past which 2 of the remaining 4 are active, and the past 2 are inactive. This tin beryllium calculated by taking the factorial of the number of spaces location are, and dividing it by the factorials of the number successful each chopped state. So successful this case, since we person 4 empty, 2 progressive and 2 inactive, we disagreement by 96.

And truthful the entropy of this is the logarithm of the number of states multiplied by the Boltzmann constant.

The adjacent earthy mobility is really the entropy evolves successful the system. It turns retired we tin make immoderate cool wide statements astir entropy based connected the topology of the graph, which I'll explicate successful a later post, and besides constitute down the conditions for which entropy increases.

Thanks to David Pfau for discussions and help, each mistakes are mine.

¹ R. Clausius, Ueber verschiedene für dice Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie, Annalen der Physik 201(7), 353–400 (1865).

² E. Schrödinger, What is Life? The Physical Aspect of the Living Cell, Cambridge University Press (1944), ch. 6.