Is the spontaneous flow of heat to thermal equilibrium an explicit law or is it implicitly assumed in thermodynamics?

Question

This sounds like a really daft question, but I am trying to clarify details on foundations on thermodynamics to myself, which will involve asking really (seemingly) basic things.

When you have two systems with a diathermal (heat and work permitting) boundary between the two systems, we know the two will spontaneously tend towards thermal equilibrium.

Is this type of statement part of any of the laws of thermodynamics or is it an implicit assumption? There are two candidates for me to look for.

• Second Law. A paraphrase of the Clausius statement says, "Heat cannot spontaneously flow from a colder location to a hotter location." This doesn't qualify, because this says what heat cannot do, it does not say what heat does do! I am looking for a positive statement of the form, "Heat spontaneously flows from a hotter location to a colder location if the two locations are separated by a diathermal boundary." The Clausius statement is not this.
• Zeroth Law. This seem like another candidate that would satisfy me, but all it says is that the relation of thermal equilibrium is a transitive relation. It doesn't say anything about the behavior of heat flow per se.

So I am wondering if the statement, "Heat spontaneously flows from a hotter location to a colder location if the two locations are separated by a diathermal boundary" is either (a) an axiom explicitly stated somewhere, (b) an assumption not explicitly stated but is used anyways, or (c) a theorem that can be derived from other laws of thermodynamics.

(Perhaps one could argue it's in the very definition of "thermal equilibrium" but I would take that to mean that (b) is the case.)

Answer 1

The phenomenon of thermal equilibration is as close as anything is to a straight up empirical law of thermodynamics. But it is also borne out neatly in the mathematical formulas of classical thermodynamics and statistical thermodynamics because of their logical consistency. The key quantity at play is entropy. If we write down Clausius' definition of entropy, \begin{gather*} dS = \frac{\delta q_{rev}}{T} \end{gather*} we can see immediately that the changes in entropy at a given temperature are directly related to the (reversible) heat flow. In the case that there is no changes in entropy, then we find that there must be no reversible flow of heat. Namely, if we are at an extremal (maximum) entropy, then there is no flow of heat, and the temperature will therefore remain fixed. In this answer and this answer, I explicitly discuss how you can construct the Clausius inequality, \begin{gather*} dS \geq \frac{\delta q}{T} \end{gather*} from the Clausius entropy and the First Law and how this applies to the transfer of heat between bodies with different temperatures. Namely, I show that the heat will always flow from the higher temperature body to the lower temperature body until they reach thermal equilibrium.

I feel compelled to also point out that the second law can be better understood as the statement that a system will tend towards thermal equilibrium by satisfying $dS \geq 0$, where the equality is satisfied at thermal equilibrium. This can be reasoned by just looking at the Gibbs entropy, \begin{gather*} S = - k_B \sum_j p_j \ln (p_j) \end{gather*} where $p_j$ is the probability of being in microstate $j$, in the limit that the system becomes isolated. This gives the Boltzmann entropy, \begin{gather*} S = k_B \ln(\Omega) \end{gather*} where $\Omega$ is the multiplicity of the microstates describing a particular macrostate or set of accessible macrostates. By just considering the probability of actually occupying the states based on their multiplicity (a set of dice can be a crude thought experiment here), we find that it is shockingly likely that if we start in some state far from the maximum entropy that the next state we occupy will be very close to the maximum entropy state. Likewise, it is incredibly likely that we will stay right there. This is the whole idea of the second law; the odds are overwhelmingly likely that entropy will constantly increase for an isolated system until we reach thermal equilibrium, at which point we will just stay there.

Answer 2

The usual account of general laws of classical thermodynamics in textbooks often does not stress presence of such law, although perhaps it should be discussed. There are formalized laws of this sort - Newton's law of cooling, or the Fourier law of heat conduction:

$$ \mathbf j = -\kappa \nabla T $$ giving heat flux density due to gradient of temperature in solids. Combined with the assumption $T$ is proportional to internal energy per unit volume, this implies the heat equation, which predicts evolution in time towards equilibrium with constant temperature everywhere.

However, these laws 1) refer to a specific linear dependence of heat flux on temperature gradient and also to the material constant $\kappa$ of the medium (thus they refer to known materials obeying this law, while not claiming it holds for all bodies; the general laws of thermodynamics do not require this law to hold in all solid media, or that $\kappa$ be non-zero) 2) are too dynamic (contain reference to time) and thus do not fit nicely into the usual formulation of classical thermodynamics which refrains (for better or worse) from using the concept of evolution in time.

The standard general laws of thermodynamics (zeroth, first, second, third) do not require that evolution towards equilibrium happens when the system is not in thermodynamic equilibrium. They only state restrictions on what processes are possible (sometimes they state which processes are not possible, like some formulations of the 2nd law). Two systems with different temperatures are not, as a whole, in thermodynamic equilibrium, and if they do not interact (e.g. if separated by a wall impermeable to heat, or removed from each other) they do not evolve towards it; and this is a valid system in classical thermodynamics.

This lack of an evolution law in classical thermodynamics does not pose a big problem when applying the laws of classical thermodynamics in practice, because 2nd law is consistent with the above heat conduction laws or even other non-linear laws, and thus general thermodynamics laws allow for many different heat-conducting, equilibrating processes, even if their exact nature and speed is not captured by any general law.

Answer 3

So the answer is, this becomes a theorem in statistical mechanics, which is our present understanding of the “stuff that's happening underneath” thermodynamics. In thermodynamics it's just a “reasonable assumption” or so, possibly having ties to the study of “ergodic” systems.

To give you a little bit of understanding about how the statistical mechanics version works,

1. In statistical mechanics systems will randomly bounce around their different microscopic configurations, as long as no conservation laws or constraints stop them. Entropy is just a measure of these microscopic configurations, the system just finds itself overwhelmingly more likely to be in higher-entropy states because there are more microscopic states there.

2. So when you say diathermal, that implies that energy can transfer between these boxes, which by (1) means it does travel both back and forward at random. But whether it will effect a net transfer depends on the increase in entropy.

3. A big energy transfer can be treated as a lot of little energy transfers. Little energy transfers can be approximated to first order.

4. So whether it will net-transfer depends on $\partial S\over\partial E$ for each of the two systems.

5. Oh wait! That's just the statistical mechanics definition of inverse temperature.

What's really happening in this argument is, I think, that statistical mechanics is forcing you to qualify the word “diathermal,” if you meant “can share energy, but might not,” then SM really needs to know the mechanism that “gates” the energy transfer in order to model the cases where it might not transfer the energy.

Answer 4

If no heat flows through the boundary, you cannot consider it "diathermal". Something must be preventing heat flow.