5 Conclusion
5.2 Meaning and implication
Unlike most researches studying N-body systems in the framework of statistical me-chanics or kinetic theory, we take a mechanical approach to a one-dimensional weakly interacting gas in thermal equilibrium. With the help from a detailed analysis of how gas particles interact, we have successfully gained some more interesting physics that the traditional approach doesn’t readily tell us. Our work is an example demonstrating the possibility to study weakly interacting N-body systems from a mechanical viewpoint working with particle trajectory.
Though the derivation here is focused on the equation of state, the spirit of this
mechanical model is investigating the physics of particle-particle interaction and trying to have a deeper understanding of a seemingly simple system. The equation of state is just a part of physics that the mechanical model can tell us. With this mechanical model, we obtain some understanding for issues other than the equation of state.
Our mechanical model consists of three elements: (1) A mechanical picture of an unperturbed system (in which there is no particle-particle interaction), (2) plausible assumptions concerning things such as the velocity distribution and homogeneity of the system that we put in by hand, and (3) assumed form of (weak) particle-particle in-teraction. For one-dimensional interacting gas at thermal equilibrium, the mechanical picture for the unperturbed system is just a bunch of particles doing constant veloc-ity motions and reversing their velocities when hitting the walls. Building mechanical model on this picture, and then put in the effect of particle-particle interaction, we capture some interesting physics. The essence of such mechanical approach lies in the fact that the effect of particle-particle interaction is weak (O (ρr0) < O (1) and O
U kBT
< O (1)), and so we can focus on two-body interaction and build on the idea of particle trajectories. To study the influence of background particles on the main particle, instead of regarding background particles as a bunch of neighbor particles surrounding the main particle, we introduce the idea of particle trajectory. For a sys-tem different from one-dimensional interacting gas (some colloidal syssys-tems or granular systems, for example), if we can figure out its free particle model and then put in particle-particle interaction, maybe we can have some interesting physics. We think it is possible to extend the idea to other systems usually studied in a statistical mechanics or kinetic theory point of view.
Statistical mechanics works with partition function. Traditional kinetic theory adopting Boltzmann equation considers a particle interacting with its neighbor par-ticles. But our mechanical approach works with a slightly more detailed analysis of the particle trajectory, the number of collisions and the effect of each collision. Here
we had restricted ourselves to the first-order calculations only, hoping that future work might extend our result. The merit of this mechanical approach is that it does provide us with more physics insight about how the individual effect affects the final answer.
Appendix A: Calculation of
12∆T
j0(without putting square well potential)
In this appendix, we calculate 12∆Tj0 starting with Eq.26. Plugging Eq.23, L1 RL 0 dx0k
Combining the second term and the third term, we have
Using the following properties
Z ∞ and the symmetry of velocity distribution, namely f (v) = f (−v), we get
1
To simplify the equation further, we utilize the feature that the interaction time is a function of the relative velocity, but not the magnitudes of the two velocities (Eq.16).
That is,
∆tinteraction(v1, v2) = ∆tinteraction(| v1− v2 |) (149) Equipped with this feature and some change of variables for Eq.148, finally, we obtain
1
2∆Tj0 = −r0N v0j + N
2 Z ∞
0
dvk0f v0k− vj0 − f vk0+ vj0 v0k v0j
2
∆tinteraction 0, vk0 . (150) Let’s stop and think about what we have done. In order to calculate ∆Tj0, we need to calculate the number of collisions and the effect of each collision. For the effect of each collision, the effect of particle-particle interaction enters through spatial displacement and time interval during the process that the main particle interacts with a background particle (Eq.17). In Section 3.1, we see that the information of displacement can be translated into the information of interaction time (Eq.13 and Eq.14), with the help of center-of-mass velocity. In other words, the effect of each particle-particle interaction is all embedded in ∆tinteraction. All the information of the form of particle-particle interaction is hidden in ∆tinteraction. As for the number of collisions, which is an idea independent of the form of particle-particle interaction, we attack this part by introducing free particle model, and the problem reduces to the counting of line-crossings of trajectories of a bunch of free particles. The validity of such free particle model relies on the fact that we are considering particle-particle interaction effect only to order one.
In deriving the above equation of ∆Tj0 (Eq.150), we have used the symmetry of velocity distribution, the assumption of spatial distribution being homogeneous, the feature that ∆tinteraction only depends on the absolute value of the relative velocity.
Notice that we haven’t put in Maxwell’s velocity distribution (so there is still f (v)) and the information of particle-particle interaction (so there is still ∆tinteraction). In other words, Eq.150 is valid for any potential that vanishes for r > r0.
Putting in Maxwell’s velocity distribution (Eq.7), we derive
1
2∆Tj0 = −r0N vj0+N
2
√ 1
aπ vj02
Z ∞ 0
dvk0
"
e−(v0k−v0j)2
a − e−(v0k+v0j)2
a
# vk02
∆tinteraction 0, v0k . (151)
Appendix B: Probability of N − 2 background particles under thermodynamic limit
In this appendix, we calculate the probability of N − 2 background particles under thermodynamic limit. We start with the probability of a single background particle in Eq.59. Defining ξ ≡ ρd3 and changing d3 to ξ NL, we note that Eq.59 becomes
probability of a single background particle
= 1
probability of a single background particle
= 1 − 1
and then rewrite Eq.153 as
probability of a single background particle
= 1 − 1
When taking the thermodynamic limit that N → ∞ and L → ∞ while NL is fixed, and the condition that ξ is fixed (that is, d3 = ξ NL is fixed under the thermodynamic limit), we have
The part dealing with the probability of the remaining N − 2 background particles is
probability of (N − 2) background particles
= (probability of a single background particle)N −2. (158)
Combining Eq.154, Eq.155, Eq.156 Eq.157 and Eq.158, and taking the thermodynamic limit with ξ being fixed, the remaining N −2 background particles part of the probability under thermodynamic limit is given by
limN →∞(probability of a single background particle)N −2
= limN →∞ 1 − 1
Appendix C: Simplification of the exponential decay probability distribution
In this appendix, we argue that the exponential decay probability distribution in Eq.95 can be replaced by unity. The basic idea is trying to show that exp
−12h
decays with d3 in a slow manner so that it can be replaced by unity. To deal with the idea of “decays in a slow manner”, we define the “decay depth”
as
That is, the decay depth is defined by
d3=decay depth
= 1
e. (161)
To simplify the decay depth, we consider its order of magnitude. Using 1 ≤h
1 + erfv0
√j
a
i
≤ 2, the order of the decay depth is given by
O (decay depth) = O
Switching to the representation by dimensionless velocity u0j = v
0
√j
a and dropping the numerical factor of√
π, we get
This is the order of magnitude of the decay depth of the probability distribution.
How about the range of d3 for non-trivial interaction we are interested in? From Fig.
17 and Fig. 20, the maximum of relevant d3 to be considered is
What is the order of magnitude of this O
maximum of relevant d3
decay depth
? If we can argue that O
maximum of relevant d3
decay depth
< O (1), then we are safe. But things are not so straight-forward. There are vk0that O
maximum of relevant d3
decay depth
is not small. For example, if v0k− v0j is quite small, that is, the velocities of the two particles are almost the same, the inter-action time will be very long and hence d3 is large in this case. This kind of vk0 do exist, but their population is quite small. They exist only in a small neighborhood around vj0, and so their population and hence their contribution to ∆vj0 is small. This is the basic idea behind the calculation and argument we will show in the following.
Now let’s check O
maximum of relevant d3
decay depth
with Eq.165. If O
maximum of relevant d3
decay depth
is not smaller than unity, then we are in trouble. Fortunately, we will see these trouble
makers actually have negligible contributions. First, we rewrite Eq.165 as the first term cause trouble? For the first term, troubles happen if
ρd2
1 + 1
u0je−(u0j)2
≥ 1, (167)
that is, we have troubles if
1 + 1
u0je−(u0j)2 ≥ 1
ρd2. (168)
Since ρd2 is small, Eq.168 means
u0j ≤ ρd2, (169)
and hence
O population u0j ≤ O (ρd2) , (170)
in view of the fact that Maxwell’s velocity distribution is √1πdue−u2 in its dimensionless form. Since the order of the population is negligible compared with the order we are willing to keep, the troubles in the first term of Eq.166 give negligible contributions.
Now let’s deal with the second term, that is
O maximum of relevant d3 To deal with this guy, we separate the situations to Case 1 and Case 2 by the order of α.
Case 1 : O (α) ≥ Oq
kBT
In this case, Eq.171 can be simplified as
O maximum of relevant d3 decay depth
= O
ρ (d2− d1)
1 + 1
u0je−(u0j)2 1 + u0j
α
. (172)
Troubles happen if
ρ (d2− d1)
1 + 1
u0je−(u0j)2 1 + u0j
α
≥ 1. (173)
Case 1-1 : If O u0j ≤ O (α), by Eq.173, troubles happen if
ρ (d2− d1)
1 + 1
u0je−(u0j)2
≥ 1, (174)
that is, we have troubles if
1 + 1
u0je−(u0j)2 ≥ 1
ρ (d2− d1). (175)
Since ρ (d2− d1) is small, Eq.175 means
u0j ≤ ρ (d2− d1) , (176)
which means
O population u0j ≤ O (ρ (d2 − d1)) . (177)
So the trouble makers do give negligible contributions.
Case 1-2 : If O u0j ≥ O (α), by Eq.173, troubles happen if
ρ (d2− d1)
1 + 1
u0je−(u0j)2u0j
α ≥ 1. (178)
That is, we have troubles if So the potential troubles still give negligible contributions.
Case 2 : O (α) ≤ Oq
kBT
In this case, Eq.171 can be simplified as
O maximum of relevant d3
BT. Troubles happen if
ρ (d2− d1)
By Eq.184 and Eq.182, the necessary condition for Case 2 to have troubles is
ρ (d2− d1)
But Eq.185 is just the same condition as Eq.173 for Case 1, which we have already checked that all the troubles give negligible contributions. So we are done.
In summary, there are some u0j, u0k that don’t respect
O maximum of relevant d3 decay depth
< O (1) . (186)
However, the violations happen for small u0j (Eq.169 and Eq.176) or the case that u0k is very close to u0j (Eq.179). The result is, the contributions given by these trouble makers are negligible in the order being considered, due to the fact that their population is small.
Appendix D: Contribution of ∆v
0j|
region−3In this appendix, we argue that the contribution of ∆vj0 |region−3 is negligible when we sum over all velocities to get the correction in momentum transferred. In view of the equation of state (Eq.11), ∆vj0 appears with the form as
X
What is the order of magnitude of this guy? Our strategy is to bound this guy and show that its order of magnitude is smaller compared with the accuracy we are retaining.
By Eq.91 and Fig. 20, the integral of d3 is bounded by
Z
And so we have
and thus is negligible. The term needed to be checked is
ρ2kBT (d2− d1)
Put Eq.192 into Eq.191, and we get
and so we are done. When putting the effect of ∆vj0 in the correction of equation of state, the contribution of ∆vj0 |region−3 is negligible, due to the fact that ∆vj0 |region−3 appears only for small vj0 and thus the population is small.
Appendix E: Particle-particle repulsion
In this appendix, we argue that particle pairs that their separations cannot reach r = d1 due to the potential barrier give negligible correction, due to the fact that their population is quite small. By Eq.128, all we need to consider is P
j
kBT , the interaction time is given by
∆tinteraction 0, v0k =
where d1 is replaced by dmin(vk0), and this is the only difference. By Eq.130, Eq.131, Eq.132 and Eq.194, the effect of replacing d1 with dmin(vk0) for ∆u < ∆u |critical in
1
2∆Tj0 is 1
2∆Tj0 |correction for ∆u<∆u|critical
= N
Expand h
e−(u0k−u0j)2 − e−(u0k+u0j)2i as
h
e−(u0k−u0j)2 − e−(u0k+u0j)2i
= e−(u0k)2e−(u0j)2
e2u0ju0k − e−2u0ju0k
= e−(u0k)2e−(u0j)2 4u0ju0k+ higher order of u0k
(196)
By Eq.196,h
e−(u0k−u0j)2 − e−(u0k+u0j)2i
contributes one order of u0kwhen u0kis small. There-fore, the order of the first term in Eq.195 is q
2Umax
kBT
3
=
2Umax
kBT
1.5
and the order of the second term in Eq.195 isq
2Umax
kBT
h−U (r)
kBT
i
= O
2Umax
kBT
1.5
. Hence the effect of replacing d1 with dmin(vk0) for ∆u < ∆u |critical in 12∆Tj0 is negligible, and so the effect is negligible in equation of state, due to the fact that the population is too small to cause relevant effect.
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