Consider a bunch of particles flying in a one-dimensional box, as shown in Fig. 2. As a particle hits the wall of the box, the wall experiences a force. The collision between the particle and the wall is assumed to be perfectly elastic. Because our description is a bit unorthodox, we first describe how we view the momentum transferred to the right-hand wall by the right-most particle in a one-dimensional box.
Physically we know the right-most particle is the only particle that is constantly bombarding the wall. But the momentum it transfers to the wall after each collision varies in time, because, prior to the collision, this particle gets its momentum from a collision with the particle immediate to its left. But where does that neighboring particle pick up that particular momentum? Clearly, It gets it through an even earlier collision with the immediate neighbor to its left. In this sense, we see that we may assign an arrow to the momentum carried by each particle, and think of each collision happening between two particles as a mechanism of exchanging the arrows of momentum. In other words, an arrow gets pushed to its right after each collision, all the way until it hits the right-hand wall.
In the picture described above, we see that the total force felt by the wall is the summation of the contributions of all particles in the box, namely
F = X
j
Fj =X
j
(momentum transferred)j
(flying time period)j =X
j
2mvj
Tj (5)
where j is the index of particle, Tj is the time period it takes for an arrow to complete one round trip (between the two walls), and vj is the velocity of a particle when it hits the wall, called collision velocity. By the lower index j, we actually label the velocity arrow, but not the particle itself, since a particle cannot pass through another particle in
L
Figure 2: Particles flying in a one-dimensional box of length L.
A
A
B
B time
v
1interaction process
v
1v
2v
2Figure 3: Interaction between particle A and particle B. Particle A is always on the left hand side of particle B, while the velocity arrow v1 can penetrate.
one-dimensional space. When two particles collide, the velocity arrows will penetrate, but the left particle stays always on the left and the right particle stays always on the right, as shown in Fig. 3. A particle is just a mediator of a velocity arrow.
For a box of ideal gas, that is, the particles have no size and there is no particle-particle interaction, the particle-particles are free, and the force is given by
F =X
j
2mv0j
Tj0 =X
j
2mvj0
2L vj0
= 2 L
X
j
1
2m vj02
(6)
Here the upper index of 0 stands for free particle. The notion of temperature is intro-duced via Maxwell’s velocity distribution
f (v) ≡ 1 q2kBT
m π
e−
1 2mv2
kB T = 1
√aπe−v2a , Z ∞
−∞
dvf (v) = 2 Z ∞
0
dvf (v) = 1 (7)
where a ≡ 2kmBT. Summation over j is given by
X
j
= N
2
Z ∞ 0
dvj0f (vj0)
= 2N
√aπ Z ∞
0
dv0je−(v0j)2
a (8)
The factor of two comes from the fact that we consider the force felt by the right wall.
Since particle hits the right wall only if its velocity is positive, we consider the right half part (positive velocity) of Maxwell’s velocity distribution and then multiply a factor of two to recover the true particle number. Plugging Eq.8 into Eq.6, the force for an ideal gas is
F = 2 L
√2N aπ
Z ∞ 0
dve−v2a 1
2mv2 = N kBT
L = ρkBT (9)
which is the one-dimensional version for P V = N kBT , the equation of state for an ideal gas. Note that Maxwell’s velocity distribution is put in by hand at the last stage just so that the population of particles with a specific velocity can be connected with temperature, provided that thermal equilibrium is reached.
What happens when particle-particle interaction is turned on? In order to study the force felt by the wall, which is equivalent to the equation of state, there are three elements: summation over all particles (P
j), flying time period (Tj), and momentum transferred (2mvj). For the summation over all the particles, we will make the explicit assumption that Maxwell’s velocity distribution is valid for our system. When particle-particle interaction is turned on, a particle-particle interacts with other particle-particles and hence its velocity is no longer a constant but influenced by particle-particle interactions. There-fore, we expect that the travel time period will be modified. For momentum transferred, due to particle-particle interaction, the velocity of a particle at the moment hitting the wall (collision velocity) may differ from its free particle velocity, and hence the mo-mentum collected by the wall should be modified in the presence of particle-particle interaction. The question we want to ask is, when there is particle-particle interaction, how do flying time period (physics in the bulk) and momentum transferred (physics on
the boundary) change, and hence lead to the modification of the equation of state? And if we stick with our mechanical approach (but with a set of extra assumptions), how far can we go? Note that standard statistical mechanics cannot tell us “what happens in the bulk” and “what happens on the boundary.” It just gives us the equation of state, the total result after combining “physics in the bulk” and “physics on the boundary.”
If the system has a high density or a long range interaction, many particles in the box are simultaneously coupled together. This makes things rather hard to track.
In order to have a working mechanical approach, we therefore consider the case of low density and particle-particle interaction being short-ranged and weak. Of course, particle-particle interaction by itself has a dimension, and we will see later that a more appropriate statement is that we consider U (r)k
BT, a dimensionless quantity, to be small.
This requirement corresponds to assuming a weak particle-particle interaction and high temperature. Since 12kBT stands for average kinetic energy, O
U (r) kBT
< O (1) means that average kinetic energy is much larger than potential energy describing particle-particle interaction. So we consider one-dimensional interacting gas systems under the limits of low density (dilute gas), short-ranged and weak particle-particle interaction (weakly interacting) and high temperature. Note that these conditions are also required by the van der Waals equation. In this sense, then, we are not imposing more conditions than the standard statistical mechanics approaches require. Since the interaction is small in some sense, we treat the effect of particle-particle interaction as a perturbation, and a particle is a free particle like ideal gas most of the time.
Compared with ideal gas, namely F = ρkBT , the correction due to particle-particle interaction in van der Waals equation of state is kept to order one, which can be seen in Eq.3. Therefore, in order to construct an equation of state from a mechanical model that can be compared with the van der Waals equation, we only need to retain the accuracy up to the first order in the particle-particle interaction. In other words, we consider the physics perturbed around the ideal gas (free particle) regime and need only
retain the perturbation up to the first order correction.
To be more concrete, our goal is to study the equation of state, which is equivalent to the force felt by the wall, and is given by
F = X
j
2mvj Tj
=X
j
2m(vj0+ ∆vj0)
Tj0+ ∆Tj0 (10)
where v0j is the free particle velocity, vj = vj0+ ∆v0j is the collision velocity, Tj0 = 2Lv0 j
is the flying time period without interaction, and Tj = Tj0 + ∆Tj0 is the flying time period in the presence of interaction. Expanding things out to first order in the small corrections, we see that Eq.10 becomes
F =X
j
2mv0j
Tj0 −X
j
2mvj0 Tj0
∆Tj0
Tj0 +X
j
2m∆v0j
Tj0 (11)
What we are going to do is trying to study the three terms in Eq.11 with the notion of particle trajectories. The first term corresponds to ideal gas, namely the unperturbed part. The second term arises from the modification of flying time period (physics in the bulk). The third term comes from the modification of collision velocity (physics on the boundary). The strategy is tracking a particle label by j (which means that its free particle velocity is vj0), called the main particle, and find out the influence of the other particles, called the background particles, on the main particle. Apparently, the main particle is not special but just a notion for bookkeeping.
Here we want to mention that the idea of temperature is actually a subtle concept.
When there is particle-particle interaction, the velocity of a particle is no longer a constant, but fluctuates as the particle interacts with other particles. How do we introduce the idea of Maxwell’s velocity distribution, and hence the temperature of the system, in this situation? The idea is, the velocity distribution of vj0 (the unperturbed part) is described by Maxwell’s velocity distribution with temperature T . That is, we identify the first term of the right hand side of Eq.11 with ρkBT , since the picture in
mind is that we are doing perturbation and things should go back to ideal gas when there is no particle-particle interaction. But is this T equal to “the real temperature of the box of interacting gas?” We don’t know. At this stage, the relation between “T ” and “the real temperature of the box of interacting gas” is unclear in our mechanical model. Later (Section 3.3) we will find out what is the real temperature of the box of interacting gas. At this stage, T is just a parameter appears inP
j (Eq.8), and we don’t know how to deal with the real temperature of the system. However, we want to point out that temperature is a thermodynamic concept instead of a mechanical concept, and hence calculations of the effect of particle-particle interaction can be done in our mechanical approach, without dealing with the thermodynamic issue of “What is the real temperature of the system?”
To avoid possible confusion, we should emphasize that we are talking about the real temperature of the system when taking standard statistical mechanics. Therefore, although we don’t know how to introduce the idea of temperature in our mechanical approach (the real temperature of the system may be larger or smaller than T in our mechanical approach), the symbol T in Eq.1 to Eq.4 is the real temperature of the system. Later we will use Trealfor the real temperature of the system in our mechanical approach. Note that if we have T 6= Treal, then we have P
j 2mvj0
Tj0 6= ρkBTreal, which means that it is too naive to identifyP
j 2mv0j
Tj0 (our mechanical approach) with the ideal gas part in Eq.1 to Eq.4 (standard statistical mechanics). Of course, T is close to Treal even if T 6= Treal, since the effect of particle-particle interaction is weak.
At the end of this chapter, we want to point out that interacting gas is different from ideal gas due to particle-particle interaction. Therefore, it is not surprising that
“how many particle-particle interactions are there” and “what is the influence of a particle-particle interaction” are at the heart of our mechanical approach when studying interacting gas. We will see how to understand the behavior of a box of interacting gas by these two central ideas.
r
U(r)
Ɛ
Ɛ>0
r=d
1r=d
2Figure 4: Square well potential.