# Is kinetic theory part of statistical mechanics?

Some years ago from now I've seem some basic details about what was then called "kinetic theory of gases" where the study of property of gases was made by statistical considerations about the momentum of the molecules and so on. One interesting thing about this is that this allowed one to view temperature as a mean of kinetic energy of the molecules.

Now last year I've studied Thermodynamics which was concerned with just macroscopic matter and right now I'm taking a course on Statistical Mechanics. When I studied Thermodynamics I thought Statistical Mechanics was all about that ideas from kinetic theory of gases but generalized to arbitrary systems.

Up to now the course just treated about the so called "microcanonical ensemble" which basically proceeded as follows: we consider a certain system and some macroscopic state described by some parameters with energy among them. Then we consider the number of microscopic states compatible with a given macroscopic energy with constant energy, that is, we give $\Omega(E)$.

From that we write down the entropy as $S(E) = k_B \ln \Omega (E)$. From this point forward, everything works like Thermdynamics, the difference is that by a counting of states we got the entropy, which on Thermodynamics was not possible to derive from anything.

Now, I don't see any relationship whatsoever between this and the kinetic theory and this made me wonder whether or not there exists a relationship between those subjects.

My point is that kinetic theory seems to provide a more detailed description based on mechanics. We have for example the Maxwell distribution of velocities telling how the velocities of the molecules distribute. On the other hand, although Statistical Mechanics as presented in the course up to now considers microscopic details of the system, it still provides just the same things Thermodynamics does, which are not so detailed on how the system behaves (for example, we know the pressure and things like that, but there's no idea whatsoever about what the molecules are doing, there are no means of dynamical quantities and so on).

So what is the true relationship between the two subjects? Are they related or they are two completely different things?

• In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time. Someone will probably correct me on this, (and I will learn same as you if they do) but to me that means if you wait long enough the energy of a system goes to a minimum...open to correction tho
– user74893
Apr 8 '15 at 18:59
• Said similarly, the ergodic hypothesis I believe is a statement that the a closed Hamiltonian system specified by a phase representative point will evolve to (or arbitrarily close to) any accessible microstate given sufficient time. It's a nice development of Liouville's theorem if I am not mistaken.
– kbh
Apr 8 '15 at 20:22
• I've decided to remove the statement about the ergodic hypothesis from the question since the real intent is to ask about how Statistical Mechanics and Kinetic Theory relate.
– Gold
Apr 14 '15 at 23:47

Based on a quick read of the Wikipedia article on the kinetic theory of gases, it looks like you would need statistical mechanics to derive any of the results in the kinetic theory of gases. For example, the Maxwell distribution of velocities is typically derived using the canonical ensemble. However, the equilibrium velocity distribution is not a dynamical quantity. If you want to get dynamical quantities like the viscosity you would have to use nonequilibrium stat mech. This is typically not taught in an intro stat mech class. If you are interested in getting dynamics from stat mech, one place to look would be Statistical Mechanics by McQuarrie. There are several chapters in that book on time correlation functions and linear response theory. It might be a bit advanced, but I think getting dynamics from stat mech is a more advanced topic than simply getting thermodynamics.

1) Statistical mechanics is more general than kinetic theory. The central object in statistical mechanics is the partition function $$Z=Tr[\exp(-\beta H)]$$ which is well defined for any system (classical or quantum) that has a Hamiltonian. The central object in kinetic theory is the distribution function $$f(x,p,t)$$ which only makes sense in the semi-classical limit. We must be able to define the notion of a probability to find a particle in phase space $(x,p)$.

2) Kinetic theory is more general than statistical mechanics. When it is applicable, kinetic theory does not require thermal equilibrium. The distribution function explicitly depends on time, and the corresponding equation of motion, the Boltzmann equation, describes the evolution of $f$ for states arbitrarily far from equilibrium. In suitable cases, the Boltzmann equation relaxes to equilibrium, and averages computed with the equilibrium distribution $$f_{eq}(x,p)$$ agree with averages computed from $Z$. Statistical mechanics has no time, and is only concerned with thermal equilibrium.

• That partition function is only valid as approximation, and it isn't the central object in statistical mechanics. It is neither true that the distribution function $f(x,p;t)$ was only valid in the "semi-classical limit". It is directly extended to the quantum domain via Wigner formulation used in quantum kinetic theory. Moreover, you are confounding statistical mechanics with equilibrium statistical mechanics; time is a fundamental parameter in non-equilibrium statistical mechanics. Jul 10 '18 at 22:44
• I think this is largely about words. Most people will use "statistical mechanics" to refer to the equilibrium theory, and "kinetic theory" to refer to a quasi-classical approximations. Of course, we can extend statistical mechanics to non-equilibrium phenomena, and kinetic theory to include quantum effects. In this case, we are talking about the same theory, non-equilibrium quantum field theory, and the distinction between statistical mechanics and kinetic theory becomes useless. Jul 20 '18 at 18:22
• Nonequilibrium quantum field theory and quantum kinetic theory are subsets of nonequilibrium statistical mechanics. Jul 22 '18 at 14:28

Kinetic theory is a subset of statistical mechanics, of non-equilibrium statistical mechanics.

Some people confounds statistical mechanics with equilibrium statistical mechanics. The latter discipline is a subset of the former.

Equilibrium statistical mechanics and thermodynamics of equilibrium (named thermostatics by Callen) only deal with equilibrium states. At equilibrium, the statistical dynamics is trivial and all what is left to analysis is the statistical part, which is conveniently described by Gibbs ensemble theory.

Gibbs theory is not valid outside equilibrium, and one has to solve the statistical equation of motion, the Liouville equation

$$\frac{\partial \rho}{\partial t} = iL \rho$$

where $L$ is the Liouvillian and $\rho=\rho(x^N,p^N;t)$ the statistical state. Solving that equation is just so difficult as solving Hamilton equations for the whole system. The goal of non-equilibrium statistical mechanics consists on developing systematic approximations to that equation for typical cases. One of those cases is the kinetic regime approximation.

Derivation of kinetic equations (Vlasov, Landau, Boltzmann,...) follows the next procedure:

1. Derive the equation of motion for the reduced one-particle state $\rho_1=\rho_1(x,p;t)$ from the Liouville equation.
2. Take the thermodynamic limit
3. Assume finite range of correlations and interactions.
4. Assume specific ordering of time and space scales $l_C \ll l_R \ll_H$ and $\tau_C \ll\tau_R \ll\tau_H$, where C means correlation, H means hydrodynamic and R means relaxation. E.g. the second set of inequalities means that typical hydrodynamic time-scale characterizing evolution of inhomogeneities has to be muchh greater than the typical relaxation time of molecular collision, which itself is muchh greater than typical time of evolution of molecule-molecule correlations.
5. Asymptotic treatment of the equation of evolution for $t \gg \tau_C$.

Deriving irreversible kinetic equations as the Boltzmann equation from a reversible equation as the Liouville equation is open to criticism. Some authors start from the Liouville equation and then add some ad-hoc extra-dynamical assumption that breaks time-reversibility and generates the correct irreversible evolutions. Other authors directly start from an irreversible extension of the Liouville equation such as the Zubarev equation

$$\frac{\partial \rho}{\partial t} = iL \rho - \epsilon \{\rho - \rho_\mathrm{ref}\}$$

In his Statistical Mechanics of Nonequilibrium Processes book Zubarev derives Boltmann equation and other kinetic equations.

Kinetics are focused on the rate and mechanism of chemical processes, so you are definitely right to say that you gain a lot of insight about mechanism from kinetics.

Many kinetic theory make extensive use of statistical thermodynamics methods, and that's why you perceive a resemblance. However, keep in mind that in kinetics the system is not in equilibrium ( in many cases far from it), so the use of statistical methods is an approximation that need some care about its applicability.

As for the statistical definition of entropy, the credit does not belong to kinetics. It is not in classical thermal dynamics but is instead from statistical thermodynamics or statistical mechanics. It is basically an extension of classical thermodynamics.

If you are interested in a level of treatment that gives insight on microscopic scale, you should be looking for statistical mechanics. Kinetics on the other hand, merely use some results from stat mech, for limits where some parts of the system can be treated as in a near equilibrium state. In many cases this can break down, especially when large amount of energy is involved. Photoreactions, for example, are often associated with breakdown of statistical models.

On the other hand , kinetics usually only pay attention to reaction rate and mechanisms in general, and does not take interest in individual details, or what happen to a molecule at a specific time. For that you'll have to turn to reaction dynamics.

• The OP is asking about the kinetic theory of gases, not kinetics. These are different things. Also, there is such a thing as nonequilibrium stat mech.
– Ian
Jun 17 '15 at 4:36