# Difference between macroscopic variable, macroscopic observable, parameter and generalized force in Thermodynamics

When I read Books about statistical physics, then often names like "macroscopic variable / observable", parameter of the macroscopic state and generalized force are used, and I want to know, what is the difference, and wether there are definitions for that. Plus, I want to know of what type are the commonly quantities V, p, N, T, S, U, $\mu$ ...

In many cases, the books I read (german literature) begin with describing a Hamiltonian System, described by the Hamilton-Function $H(\Gamma)$ on the Phase-Space, which has so many dimensions, that even numerical calculations are not possible, and thus one searches for another way of describing the system, by giving Distributions and calculating mean-values.

In the Books that I worked with, it is stated that the Hamilton-Function can additionaly depend on external Parameters, which affect the energy of the microstates (one microstate is one point in the phase space of the system).

Question 1: Is N such a parameter? I can't imagine how the Hamilton-Function of a classical System can depend on N in a numerical way in which N denotes the Number of particles in the system. The same goes for V: How can the Hamilton-Function depend on one number V ? I could imagine that the Hamilton-Function contains some external Potential-pot, whose shape could depend on some variables.

Afterwards: Observables: I read about Observables, that are mean values of functions on the phase space: For every function $O(\Gamma)$ I can calculate and measure $<O> = \int d\Gamma \rho(\Gamma) * O(\Gamma)$.

Question 2: For Systems that don't have a sharp volume, like in the pressure Ensemble, the volume is given as a mean value, dependend on the distribution of the microstates of the system. But I can't imagine something like a "Volume-function" on the Phase-Space. How do I measure that? Is the Volume of a microstate the volume of the smallest region that contains every particle of the system? If so, what shape does it have? Is it a square? I could think of abitrary possibilities to asign a volume to a given set of coordinates of particles, and that is my problem.

Next, the concept of generalized forces was introduced to me as the mean-value of derivations of the Hamiltonian to the external Parameters, so for every external Parameter $X_i$ there is a corresponding force $F_i = \int d \Gamma \rho (\Gamma) \frac{\partial H}{\partial X_i}$.

Question 3: Is $\mu$ a generalized force, like the ones described above? I can't imagine how I should derivate H after a Number of particles, because, as stated in question 1, I don't know how H depends on N in a differentiable way.

Further Questions:

Can I calculate a mean-value of the system for every parameter of the system? Which of those quantities is entropie? It's not a mean-value, but it also isn't a parameter in the hamilton-function, as opposed to the energy U, which is the mean-value of the hamilton-function.

Last question(still important): How do I decide which of those macroscopic quantities are natural variables of the system, and which of them are thermodynamic potentials? Is there any convention?

## 1 Answer

The general principle is that macroscopic variables and macrostates are not "real" from the microscopic, Hamiltonian perspective. They're things that we, human beings on the scale of $10^{23}$ atoms, make up based on what we can observe.

For example, let's take pressure. Given a microstate $\Gamma$, you can't calculate the pressure, because such a thing doesn't exist. When we measure a pressure, we are really measuring the time average of a force over a (microscopically) long time. The microstate knows nothing about this.

Said another way, the ergodic hypothesis says a time average is an ensemble average, so a pressure reading is an ensemble average. Therefore, it makes no sense to talk about the pressure of a single member of an ensemble.

Similarly, entropy is macroscopically defined. There is no way to calculate "the entropy of a microstate" because entropy is the amount a macroscopic observer doesn't know about an ensemble of microstates.

You might be confused because all of the above discussion falls apart if you consider systems of only a few particles. And that makes sense, because these macroscopic, thermodynamic variables are not defined for such systems. Thermodynamics is the $N \to \infty$ limit. To apply it, you use real-life, $N > 10^{23}$ intuition.

• I'm only answering the first ~2 questions here; 5 separate questions is too much for one thread. Commented Jan 10, 2016 at 21:53
• I am aware that most of the macroscopic variables can't be calculated for a microstate. But still, given a distribution $\rho(\Gamma)$ (which for me is the same thing as a macrostate), I somehow have to asign this distribution the macroscopic variables that, as you said, are maid up. For example, you can calculate entropy when you know the distribution $\rho$, then $S = - k_B \int d\Gamma \rho ( \Gamma ) Ln ( \rho ( \Gamma))$. Isn't it that in the same way, you have to define any macroscopic value? Otherwise you can't formulate laws for those values derived from statistical physics? Commented Jan 10, 2016 at 23:08
• Right, and I'm saying that if you want to calculate pressure/volume/etc. this way, you can't. People sweep the issue under the rug. For example, for the pressure ensemble (which I think is also called the Gibbs ensemble), the distribution $\rho(\Gamma)$ is proportional to $\exp(-\beta (H(\Gamma) + pV(\Gamma)))$. Commented Jan 10, 2016 at 23:21
• Here, the volume $V(\Gamma)$ is just put in by hand. You can't derive it from $H(\Gamma)$, which is a totally separate quantity. For example, for a gas under a piston, $V$ comes from the location of the piston, but $H$ only depends on the positions/momenta of the particles. Commented Jan 10, 2016 at 23:23
• What do you mean, $V(\Gamma)$ is put in by Hand? I'm not seeking to derive it from the Hamiltonian, but the question still is, by what way do you asign the Volume $V(\Gamma)$ to a microstate $\rho$? In your example, the location of the piston does not depend on the microstate, although you write $V(\Gamma)$ Commented Jan 11, 2016 at 0:04