Statistical mechanics and kinetic theory are two main frameworks one adopts when studying a system containing a huge number of degrees of freedom. Researches in this field study hard sphere systems [2, 12, 14, 15, 16, 17, 19, 20], hard ellipse systems [4], hard needle systems [6, 7], van der Waals theory [8], granular particle systems [1, 5, 11], etc. There are also researches concerning fundamental issues [9, 10].
Statistical mechanics and kinetic theory are powerful and systematic, but it often amounts to meaning that one has to pay the price of losing certain detailed dynam-ical information about the interparticular interactions. In order to study a weakly interacting N-body system, besides statistical mechanics, kinetic theory and numerical simulation, do we have other choice? At first glance, the idea of particle trajectory is useful only for systems containing small degrees of freedom. However, we can actually bring in the idea of trajectory for a N-body system if the effect of particle-particle interaction is weak. In this work, we study a weakly interacting one-dimensional gas to see how a direct approach investigating the detailed mechanical interaction between particles might shed some light on its merits as opposed to the traditional approach.
Before describing our mechanical approach, we first give a brief review of how one understands a weakly interacting one-dimensional gas in statistical mechanics. For a one-dimensional classical gas in the box, the equation of state at thermal equilibrium can be calculated by standard recipes in statistical mechanics. Specifically, one can start with a given Hamiltonian of the gas system and computes its partition function, then following through the standard recipes [3, 13] to arrive at the following equation of state
F = ρ(kBT ) − ρ2(kBT ) Z ∞
0
dr
e−
U (r) kB T − 1
+ O(ρ3), (1)
where F is the force exerted on each side of the confining box, ρ ≡ NL is the linear number density (N and L are the particle number and the size of the one-dimensional
r
U(r) r=d
1Figure 1: Particle-particle interaction consisting of a hard core (at r = d1) and an interaction tail. U (r) is the potential describing the interaction between two particles.
box, respectively), kB is Boltzmann’s constant, T is the temperature of the system, and U (r) is the potential between two gas particles with r being their separation. In the above, one has assumed a low density limit so that a Taylor expansion in ρ is possible.
We will specialize to the case when the potential representing particle-particle in-teraction is consisted of a hard core and an inin-teraction tail, as shown in Fig. 1.
By Eq.1, the equation of state for such a potential is given by
F = ρ(kBT ) + ρ2(kBT )d1− ρ2(kBT ) Z ∞
d1
dr
e−
U (r) kB T − 1
+ O(ρ3). (2)
Consider only the low density limit and keep up to the second order in the number density, and further assume the high temperature limit and weakly interacting limit.
Approximate Eq.2 to first order of U (r)k
BT under these limits, and the equation of state becomes
F = ρ(kBT )+ρ2(kBT )d1−ρ2 Z ∞
d1
dr [−U (r)] = ρ(kBT )
1 + ρd1− ρ Z ∞
d1
dr −U (r) kBT
(3)
After rearrangement, the equation of state is given by
F + (N L)2
Z ∞ d1
dr [−U (r)]
(L − d1N ) = N kBT, (4)
which is the van der Waals equation in one-dimension.
This is the statistical mechanics approach to the van der Waals equation, the equa-tion of state for interacting gas under the condiequa-tions of low density, high temperature and weak interaction. The derivation provided by statistical mechanics is simple, and in one scoop it easily relates the force with the temperature, the density and the po-tential describing particle-particle interaction. On the other hand, one also loses track of exactly what has happened to the dynamical behavior of the constituent particles.
For example, if we “turn on” the interparticular interactions of an otherwise ideal gas, will that make the gas particle move faster or slower? And, does the answer depend on the original velocity of a particle? Statistical mechanics by itself doesn’t provide such information. In this regard, the physics contained in an equation of state is quite limited.
In this work, we adopt an approach different from statistical mechanics. We take a mechanical point of view, considering the detailed mechanics of the gas particles.
In order to deal with a system of an essentially infinite number of degrees of freedom, statistical mechanics hides the complexity in its fundamental principles and assumptions (such as the notion of partition function), so that we can focus on things like the equation of state that can be more readily measured. In our mechanical approach, in order to deal with the complex one-dimensional interacting gas system, we still need to put in by hand some basic assumptions. We are not pretending that we are going to derive the equation of state for a one-dimensional interacting gas from scratch and nothing else. However, we will do our best making assumptions that seem at least reasonable. Besides rebuilding the equation of state, we will have some physics that standard statistical mechanics doesn’t tell us. For example, we will try to answer the
two previous questions in our mechanical approach.
The thesis is organized as follows. In Chapter 2, we discuss the foundation of our mechanical point of view. In Chapter 3, we consider a special case of square well particle-particle interaction and show how the mechanical model is built. In Chapter 4, we generalize the previous results to generic interacting gas, with the help of physics insight we developed in Chapter 3. Finally, we summarize the result and give some possible implications in Chapter 5.