# 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?

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. 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? 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 $\exp(-\beta (H(\Gamma) + pV(\Gamma)))$. 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. 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 $V(\Gamma)$ Jan 11, 2016 at 0:04