Characteristics of the transition probability

In this section some exact properties of the transition probability introduced in the previous section are derived from Eq. (2.18). The approximation to this probability that is going to be built at the end of this chapter will be constructed in such a way that it obeys all of these properties.

2.2.1. Detailed balance

Since the underlying model is microscopically reversible, detailed balance holds for the Markov process: the transition probability for the process E_tr → E_tr⁰ =E_tr² is related to the transition probability for the reverse process E_tr⁰ →E_tr=E_tr⁰ /² via

p_T⁰

B(E_tr →E_tr⁰ )e^−Etr/TB0 =p_T⁰

B(E_tr⁰ →E_tr)e^{−Etr0 /TB0}. (2.25) Here fluctuations of the temperatures of the baths have been neglected which seems justified because changes in the bath temperature are O(1/N_i^mod) (see Eq. (2.24)).

2.2. Characteristics of the transition probability

Detailed balance, Eq. (2.25), implies the following relation for p_β():

p_β()e^−β =p²_β 1

e^−2β. (2.26)

Since detailed balance is a property of the equilibrium state of the two particle system, this is strictly valid only if the temperatures of both rods are equal. Later, for lack of a better scheme, it will also be used for rods with different bath temperatures.

2.2.2. High and low temperature limits

When the temperatures of the baths of oscillators are zero, i.e. q(τ) = 0 for all τ, the collision is deterministic. For this case, Eq. (2.18) can be solved exactly

“by inspection”: As long as the expression inside the curly braces in Eq. (2.18) is monotonically increasing, which it is for τ⁰ < τ₀ + Γ₁ (recall that Γ₁ < Γ₂), the inner max-function is redundant such that w(τ) = (τ−τ0)Θ(τ−τ0) for τ < τ0+ Γ1. For largerτ, the memory terms compensate all the gain from the term τ −τ₀, thus the final value of w(τ) is Γ₁. Hence it follows that the coefficient of restitution is = Γ1 −1 = γ, the length ratio of the rods. This is well known and has been shown e.g. in [Aue94, GZ96]. Apparently, the earliest derivation of this result seems to be due to Saint-Venant as early as 1867, according to the reference on p. 283 in Rayleigh’s article [Ray06]³. Thus in the limit of small temperatures, pβ() should approach a δ-Function centered around γ.

At large temperatures, on the other hand, p_β() is a uniform distribution, i.e.

pβ=0() = const. (within a certain range and zero outside of it). This is proved in App. A.4.

2.2.3. Maximum collision time

If the length ratio γ is a rational number, it is possible to extract some more infor-mation aboutpβ() from Eq. (2.18). First one needs to know the maximum duration of a collision before an upper and lower bound for can be deduced.

The stochastic process q(τ) consists of two periodic Brownian bridge processes with periods Γ1 and Γ2 respectively. If γ is rational, sayγ =p/s (where the integers pand s are relatively prime), then Γ₁/Γ₂ =p/sis rational and thus q(τ) is periodic, the period being given byp+s. Letτ^∗ be the time whereτ−τ₀+q(τ) first equals 0 and is greater than 0 in an (arbitrarily small) interval to the right ofτ^∗ (see Fig. 2.1).

In other words, τ^∗ is the time when the particles first touch and the collision begins.

The periodicity of q(τ) now leads to a maximum collision time because whatever

3A slightly more involved calculation than the one mentioned here shows that amazingly w(τ) assumes the same final value Γ1even if only the longer rod is nonvibrating while the shorter one vibrates with arbitrary strength.

2. One-dimensional particles 1: The two particle system stochastic process q(τ) is periodic with period 3. The memory terms (the dotted and short dashed curves) are attached to the curve forw(τ) for illustration. The timeτ^∗ marks the beginning of the collision.

w(τ) is for a particular realisation ofq(τ), the memory terms which accumulate over-compensate the gain from the termτ⁰−τ₀+q(τ⁰) after timeτ^∗+p+s(see Eq. (2.18) and Fig. 2.1 for illustration).

This can be seen more clearly by the following argument. By evaluating Eq. (2.18) at τ =τ^∗ +p+s it is clear that

The last step follows from the definition of τ^∗ and the periodicity of q(τ). When the expression inside the curly braces in Eq. (2.18) is rewritten in the following way by splitting the double sum in three parts, one for i = 1 and ν ≤ s, the second one for i = 2 and ν ≤ p, and the third one containing the rest (keeping in mind that

2.2. Characteristics of the transition probability the last four terms can be estimated by use of Ineq. (2.27): p+s−Ps

ν=1w(τ⁰−νΓ₁)− Pp

ν=1w(τ⁰−νΓ₂)≤w, first only for˜ τ⁰ =τ^∗+p+s but sincew(τ) is a monotonically increasing function, this also applies for all τ⁰ > τ^∗+p+s. From the two sums the last terms could even be dropped because they are zero forτ⁰ =τ^∗+p+s. However, only one of these last terms will be dropped, resulting in

p+s− lhs of Ineq. (2.30) was just the expression in curly braces from Eq. (2.18) and ˜w = w(τ^∗ +p+s), this means that no further contributions to w(τ) come from times τ > τ^∗ +p+s and it therefore follows that p+s is the maximum collision time.

2.2.4. Upper and lower bound for

Since the maximum collision time isp+s, we have=w(τ^∗+p+s)−1 (cf. Eq. (2.22)).

Bounds on can now be derived from Eq. (2.18), evaluated at τ = τ^∗ +p+s, by neglecting or over-estimating the memory terms.

2. One-dimensional particles 1: The two particle system

Consider first Eq. (2.18) atτ =τ^∗+p+s with all memory terms neglected which yields the following inequality:

w(τ^∗+p+s)≤max

0, max

τ⁰∈[0,τ^∗+p+s]{τ⁰ −τ0+q(τ⁰)}

. (2.31)

Since τ^∗ is the beginning of the collision, it is clear that τ⁰ −τ₀ +q(τ⁰) ≤ 0 for all τ⁰ ≤τ^∗. Thus it is also clear thatτ⁰−τ₀+q(τ⁰)≤p+s for allτ⁰ ≤τ^∗+p+s(simply by replacingτ⁰ byτ⁰−p−sand using the periodicity ofq(τ)). At timeτ⁰ =τ^∗+p+s, this inequality even becomes an equality. Therefore, the rhs of Eq. (2.31) is equal to p+s, so one gets

+ 1 =w(τ^∗+p+s)≤p+s. (2.32) Next, the at mostp+s−2 non-zero memory terms in Eq. (2.18) at timeτ^∗+p+s are over-estimated by the maximum possible value, namely w(τ^∗+p+s) =+ 1:

w(τ^∗+p+s)≥max

0, max

τ⁰∈[0,τ^∗+p+s]{τ⁰−τ₀−(p+s−2)(+ 1) +q(τ⁰)}

. (2.33) By the same argument as above, the rhs of Eq. (2.33) is equal top+s−(p+s−2)(+1).

Thus one has

+ 1 =w(τ^∗+p+s) ≥ p+s−(p+s−2)(+ 1)

⇔ ≥ 1

p+s−1 (2.34)

It is easy to see that the upper and lower bounds on are also compatible with the detailed balance condition, Eq. (2.26), as it should be.

Because it will be needed later on, the parameter ξ is introduced here for conve-nience:

ξ = 1

p+s−1. (2.35)

The bounds calculated in this section are optimal bounds. This follows from the fact that p_β=0() is a uniform distribution between ξ and 1/ξ which is proved in App. A.4.

2.2.5. The special case γ = 1

When the length ratio of the rods, γ, is equal to 1, an exact solution forp_β() can be deduced from the results of the preceding sections. If γ = 1, the periods of the two