GoKawiilClausius (1865)¹ defines a quantity called entropy. By decomposing physical processes as a chain of engines, he shows that entropy always increases for irreversible processes. For reversible processes like Carnot's ideal engine, the change in entropy is zero. But when an irreversible process occurs, entropy can never decrease unless energy is applied to a system. This is what is known as the second law of thermodynamics.
And whilst entropy itself may not be measurable with a thermometer or ruler, it is still a useful concept since we can calculate derived quantities from it that are directly measurable.
I've written previously quite vaguely about 'life as entropy'. This was an idea motivated by Schrödinger (1944)² through the concept of negentropy. Through negentropy, life seems to maintain order by feeding on the energy around it, and reducing its local disorder. But up until now, I've been confused about what this actually means in detail.
And so one way I'm trying to understand this is through models. One approach might be to make toy models and then figure out a mathematically consistent way to define entropy in those systems. Then, you could simulate the model and have a better idea of how the entropy evolves.
One such toy model that is useful to get an intuition is Dyson's toy model of a cell. Dyson's toy model of a cell is a Markov chain that settles into one of three equilibrium states, two of which are the 'life' and 'death' states. But if we care about the vague concept of 'life as entropy', then we should be able to make a definition of entropy to apply to the Markov chain.
But the concept of entropy as originally defined by Clausius was a function of work and temperature
\(\mathrm{d}S \;=\; \frac{\delta Q_{\text{rev}}}{T}\)
And so it is unclear how to relate the two ideas together. Is there a different way to define entropy in the context of Markov chains that gives you the same result, consistent with physical ideas?
Well, one way to attack this is by looking at the work of Boltzmann, who quantified the relationship between entropy and the number of possible states in a system.
Suppose you take a system and then observe a system's state variables, like temperature, pressure and volume, using various instruments. Presumably, there are a number of different configurations of the physical system that are compatible with it being in that state. And if entropy is related to the concept of order and disorder, then presumably entropy would be higher if there were more possible states.
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