Now I know there are a few questions related to mine (here and here), but none of them (and their answers) really touch what I don't understand in Section 7 (Entropy) of Statistical Physics from Landau & Lifshitz (I use the first edition). This is not my first time delving into statistical physics. I actually have a PhD in physics but I use only rather basic thermodynamic relationships and my undergrad days were a long time ago, I forgot a lot of this stuff. I really enjoyed the first two L&L books so I wanted to dive into this one. In addition, I would like to have opinions on an interpretation of mine of the second law of thermodynamics.
In Section 6, L&L define "the probability that the system is in one of the states $\mathrm{d}\Gamma$" as $$\mathrm{d}w=\mathrm{const.}\times\delta(E-E_0)\Pi_a\mathrm{d}\Gamma_a,$$ where the index $a$ refers to the subsystems making up the closed system of interest, $\mathrm{d}\Gamma=\Pi_a\mathrm{d}\Gamma_a$ and $E_0=\sum_aE_a$ is the total energy. Now, a larger probability that the system finds itself in the range of states $\mathrm{d}\Gamma$ of course means that the latter is larger. The more states there are in that interval, the more probable it is that the system is in one of those states.
Now in Section 7, they define the probability that the energy of a subsystem lies between $E$ and $E+\mathrm{d}E$ as $W(E)\mathrm{d}E$. With $\Gamma(E)$ the number of states with energies less than or equal to $E$, the number of states with energies between $E$ and $E+\mathrm{d}E$ is $$\frac{\mathrm{d}\Gamma(E)}{\mathrm{d}E}\mathrm{d}E.$$ The energy density of the states is then $\mathrm{d}\Gamma(E)/\mathrm{d}E$. From whis we find that the probability density (per energy) is given by $$W(E)=\frac{\mathrm{d}\Gamma(E)}{\mathrm{d}E}w(E),$$ $w(E)$ being the probability density (per state), which is itself a function of energy. Furthermore, the normalization condition $\int W(E)\mathrm{d}E=1$ must hold. Since statistical fluctuations are extremely small for macroscopic bodies, the above equation becomes $W(\overline{E})\Delta E=1$, or $w(\overline{E})\Delta\Gamma=1$, where $$\Delta\Gamma=\frac{\mathrm{d}\Gamma(\overline{E})}{\mathrm{d}E}\Delta E$$ and $\overline{E}$ is the mean energy of the subsystem, with the statistical fluctuations $\Delta E$ being very small. L&L then define the entropy as $S=\log\Delta\Gamma$.
Now this is where it gets tricky for me. They rewrite $\mathrm{d}w$ as $$\mathrm{d}w=\mathrm{const.}\times\delta(E-E_0)\Pi_a\frac{\mathrm{d}\Gamma_a}{\mathrm{d}E_a}\mathrm{d}E_a.$$ So now the probability that the system is in one of the states $\mathrm{d}\Gamma$ is given in terms of the energy density of the states and the associated range of energies. They then write:
The statistical weight $\Delta\Gamma_a$ is by definition a function of the mean energy $\overline{E}_a$ of the subsystem, and so is the entropy $S_a=S_a(\overline{E}_a)$. We may formally regard $\Delta\Gamma_a$ and $S_a$ as functions of the actual value of the energy $E_a$ (the same functions which they actually are of $\overline{E}_a$). Then we may replace the derivatives $\mathrm{d}\Gamma_a(E_a)/\mathrm{d}E_a$ by the ratios $\Delta\Gamma_a/\Delta E_a$, where $\Delta\Gamma_a$ is understood in the above sense as a function of $E_a$ and $\Delta E_a$ is the energy interval corresponding to $\Delta\Gamma_a$ (also a function of $E_a$).
After some steps, then find the expression $$\mathrm{d}w=\mathrm{const.}\times\delta(E-E_0)e^S\Pi_a\mathrm{d}E_a.$$ Following this equation, L&L write:
We know the most probable values of the energies $E_a$ are their mean values $\overline{E}_a$. This means that the function $S(E_1, E_2,...)$ will have its maximum possible value (subject to $\sum E_a=E_0$) at $E_a=\overline{E}_a$. But the $\overline{E}_a$ are just those values of the energies of the subsystems which correspond to the state of total statistical equilibrium of the system. Thus we arrive at the following most important deduction. The entropy of a closed system has its greatest value (for a given value of the energy of the system) in a state of total equilibrium.
From this deduction, it is pretty straightforward to arrive at the principle of maximum entropy.
Now here are my questions:
- Since $S_a$ and $\Delta\Gamma_a$ are defined in terms of the mean energy $\overline{E}_a$ of the subsystem, I do not understand why we can just write them as functions of the actual energy. By definition, $\Delta\Gamma_a$ is related to $\Delta E_a$, the statistical fluctuation around the mean energy $\overline{E}_a$. $\Delta\Gamma_a$ then represents the different states over this energy fluctuation. What then, does $\Delta E_a$ mean in the context of the actual energy of the system?
- This leads to my second question. The entropy is defined in terms of the range of states corresponding to a narrow spread of energy due to statistical fluctuations around the mean energy at statistical equilibrium. This spread in energy corresponds to different macrostates, with each their own microstates. Now, from what I remembered, entropy is related to the total number of possible microstates. How does this relate to a spread of macrostates?
I suppose answering these questions would help me understand the last equation. From it, it is obvious that larger values of $S$ implies larger probabilities. But I do not understand how this $S$ relates to any value of the energy. In other words, from this derivation, I do not understand why a macroscopic body in thermal equilibrium has more available states than one not in thermal equilibrium. And are we talking about the total number of states, or a range of states?
In order to explain that entropy always increases, we often see the example of the box divided by two, one half is empty and the other half contains a certain number of particles. By removing the wall, the different number of positions for the particles increases. I never really liked this explanation because I feel it is too specific. What separates a state of statistical equilibrium to one not in statistical equilibrium? The presence of a physical process going on. Physical kinetics, the physics of non-equilibrium systems, in particular deals with determinations of molecular viscosity, magnetic diffusivity or thermal conductivity. Each of these are related to processes which are of course dissipative and leads to an increase of entropy. But take viscous dissipation for example. It is usually seen as the diffusion of vorticity down its gradient. It involves a direction.
In these processes, the fluid particles are not free to move however they want (in a way consistent with intermolecular collisions of course). They are constrained in their movement, towards a certain direction, due to the physical process at hand. In a way, these processes exclude certain microstates by demanding a certain result. When these processes are over, the constraints are lifted and microstates are no longer excluded.
Now this is not true for all physical processes. Advection is one example. Far from boundaries, you can always find a macroscopic part of the flow for which, in a certain given time, the flow is steady and the fluid is in local thermodynamic equilibrium. The state of the system now no longer only depend on energy, but also on the linear momentum. But this parcel of fluid remains in thermodynamic equilibrium. Although I suppose there could be extreme examples where the flow varies in space and/or time too quickly compared to the relaxation time of the smallest possible macroscopic parcels of fluids. Gravity at the supposed highly homogeneous Big Bang conditions would be another such source of directionality or anisotropy.
So does this explanation makes sense?
