Lanford’s Theorem and the Emergence of Irreversibility

Abstract

It has been a longstanding problem to show how the irreversible behaviour of macroscopic systems can be reconciled with the time-reversal invariance of these same systems when considered from a microscopic point of view. A result by Lanford (Dynamical systems, theory and applications, 1975, Asterisque 40:117–137, 1976, Physica 106A:70–76, 1981) shows that, under certain conditions, the famous Boltzmann equation, describing the irreversible behaviour of a dilute gas, can be obtained from the time-reversal invariant Hamiltonian equations of motion for the hard spheres model. Here, we examine how and in what sense Lanford’s theorem succeeds in deriving this remarkable result. Many authors have expressed different views on the question which of the ingredients in Lanford’s theorem is responsible for the emergence of irreversibility. We claim that these interpretations miss the target. In fact, we argue that there is no time-asymmetric ingredient at all.

Appendix

In this appendix we consider in more detail the issue of time-reversal invariance for the Boltzmann equation and two versions of the BBGKY hierarchy equations and prove the claims concerning this issue in the body of the paper. The commonly accepted criterium for judging time-reversal invariance of such evolution equations, describing a density function (either $f$ or $\rho _k$) over particle configurations is as follows: the equation determines a class $S$ of allowed solutions, where each solution can be seen as a ‘history’ of the density function, either $\mathcal{H}:= \{ f_t, \; t\in {\mathbb {R}}\} \in S$ or $\mathcal{H} := \{ \rho _{k,t}, \; t\in {\mathbb {R}}\}$, carving out, so to say, a trajectory in their respective spaces of all density functions that solve the evolution equation.

Now consider the time-reversal of such a history, defined as $\mathcal{TH} := \{\tilde{f}_{t} \; t\in {\mathbb {R}}\}$ or $\mathcal{TH}:= \{\tilde{\rho }_{k,t} \; t\in {\mathbb {R}}\}$ , where $\tilde{f}_t (\vec {q}, \vec {p})= f_{-t} (\vec {q},- \vec {p})$ and $\tilde{\rho }_{k,t} (\vec {q}_1, \vec {p}_1; \ldots ; \vec {q}_k , \vec {p}_k) = \rho _{k,-t} (\vec {q}_1, -\vec {p}_1; \ldots ; \vec {q}_k ,- \vec {p}_k)$ are obtained from $f_t$ and $\rho _{k,t}$ by reversing $t$ and the momenta in their arguments. The question is then whether such a time-reversed history $\mathcal{TH}$ is a solution of the equation too, whenever $H$ is a solution of the equation in question. In other words: Is $\mathcal{T} H \in S$ whenever $T \in S$? If the answer is yes, the equation is time-reversal invariant, otherwise not.

Our strategy will be the same in all three cases. We consider an arbitrary solution of the equation and construct from this a time-reversed history, and derive the equation it obeys. If the resulting equation is equivalent to the original equation we have proved that the original equation is time-reversal invariant. But if it is not, the original equation is not time-reversal invariant. To be sure, our first two propositions below are not surprising: every author in the field recognizes that the Boltzmann equation is not time-reversal invariant, whereas the BBGKY hierarchy is, even if detailed proofs of these assertions are hard to find. We have added these proofs mainly to show that that by the same standard of rigour, one can obtain Proposition 3, which we believe to be of more interest.

Proposition 1

The Boltzmann equation (8) is not time-reversal invariant.

Proof

Since nothing interesting happens to the position variables in the Boltzmann equation (8) we will suppress them in the notation below, and also put the mass $m=1$. Note that $\vec {p}_1$ is the only independent momentum variable in the equation: $\vec {p}_2$ appears only in the right-hand side as a mere integration variable and the outgoing momenta variables $\vec {p}_1^{\ \prime }, \vec {p}_2^{\ \prime }$ in the collision integral are functions of $\vec {p}_i$: that is, $\vec {p}_i^{\ \prime } =\vec {p}_i^{\ \prime }(\vec {p}_1, \vec {p}_2) = T_{\omega _{12}} (\vec {p}_1, \vec {p}_2)$, where $T_{\omega _{12}}$ is defined by (7). So, let $f_t (\vec {q}, \vec {p}) , t\in {\mathbb {R}}$, by an arbitrary solution of the Boltzmann equation (8), and consider the joint transformation of $t\longrightarrow -t$ and $\vec {p}_1 \longrightarrow - \vec {p}_1$. Applying this transformation to the left-hand side of (8) leads to:

$$\begin{aligned} - \displaystyle \frac{\partial }{\partial t} {f}_{-t}(- \vec {p}_{1}) \displaystyle -{\vec {p}_{1}} \cdot \frac{\partial }{\partial \vec {q}} \,{f}_{-t}(- \vec {p}_{1}) = \displaystyle \frac{\partial }{\partial t} \tilde{f}_{t}( \vec {p}_{1}) \displaystyle +{\vec {p}_{1}} \cdot \frac{\partial }{\partial \vec {q}} \,\tilde{f}_{t}( \vec {p}_{1}), \end{aligned}$$

(56)

while the right-hand side of (8) becomes

$$\begin{aligned}&Na^{2} \displaystyle \int \! d \vec {p}_2 \int _{\vec {\omega }_{12} \cdot (\vec {p}_{1} +\vec {p}_{2}) \leqslant 0} \! d \vec {\omega }_{12} \,(-\vec {p}_{1} - \vec {p}_{2}) \cdot \vec {\omega }_{12} \,\nonumber \\&\quad \left[ {f}_{-t}( \vec {p}_{1}^{\prime }) {f}_{-t}( \vec {p}_{2}^{\prime } )-{f}_{-t}(-\vec {p}_{1})f_{-t}(\vec {p}_2) \right] \end{aligned}$$

(57)

Here, the notation $\vec {p}_i^{\ \prime \prime }$ is used to indicate that these momenta have to be thought of as functions of $(-\vec {p}_1, \vec {p}_2)$: $(\vec {p}_1^{\ \prime \prime }, \vec {p}_2^{\ \prime \prime }) = T_{\omega _{12}} (-\vec {p}_1, \vec {p}_2)$.

If we now perform an additional (cosmetic) transformation of the integration variables $\vec {p}_2 \longrightarrow - \vec {p}_2$ and $\vec {\omega }_{12} \longrightarrow - \vec {\omega }_{12}$, (57) can also be written as

$$\begin{aligned}&Na^{2} \displaystyle \int \! d \vec {p}_2 \int _{\vec {\omega }_{12} \cdot (\vec {p}_{1} - \vec {p}_{2}) \geqslant 0} \! d \vec {\omega }_{12} \,(\vec {p}_{1} - \vec {p}_{2}) \cdot \vec {\omega }_{12} \,\nonumber \\&\quad \left[ {f}_{t}( \vec {p}_{1}^{\ \prime \prime \prime }) {f}_{t}( \vec {p}_{2}^{\ \prime \prime \prime }) -{f}_{t}(-\vec {p}_{1}) {f}_{t}(- \vec {p}_{2}) \right] , \end{aligned}$$

(58)

where $(\vec {p}_1^{\ \prime \prime \prime }, \vec {p}_2^{\ \prime \prime \prime }) := T_{\omega _{12}} (-\vec {p}_1, -\vec {p}_2)$. But $T_{\omega _{12}}$ is a linear operator, and therefore $\vec {p}_i^{\ \prime \prime \prime } = -\vec {p}_i^{\ \prime }$. If we now substitute $f_{-t}(-\vec {p}) = \tilde{f}_t( \vec {p})$ in (58) and equate the transformed left-hand side (55) of (8) to the transformed right-hand side (58) of (8), we find that $\tilde{f}_t$ satisfies the equation

$$\begin{aligned}&\displaystyle \frac{\partial }{\partial t} \tilde{f}_{t}( \vec {p}_{1}) \displaystyle +{\vec {p}_{1}} \cdot \frac{\partial }{\partial \vec {q}} \,\tilde{f}_{t}( \vec {p}_{1}) = \nonumber \\&-Na^{2} \displaystyle \int \! d \vec {p}_2 \int _{\vec {\omega }_{12} \cdot (\vec {p}_{1} - \vec {p}_{2}) \geqslant 0} \! d \vec {\omega }_{12} \,(\vec {p}_{1} - \vec {p}_{2}) \cdot \vec {\omega }_{12} \, \left[ \tilde{f}_{t}( \vec {p}_{1}^{\prime }) \tilde{f}_{t}( \vec {p}_{2}^{\prime }) \!-\!\tilde{f}_{t}(\vec {p}_{1}) \tilde{f}_{t}(\vec {p}_{2}) \right] ,\nonumber \\ \end{aligned}$$

(59)

also known as the anti-Boltzmann equation. We conclude: whenever the solution $\{ f_t \;, t\in {\mathbb {R}}\}$ satisfies the Boltzmann equation, the time reversed solution $\{ \bar{f}_{-t} \;, t\in {\mathbb {R}}\}$ solves the (inequivalent) anti-Boltzmann equation, and therefore, the Boltzmann equation is not time-reversal invariant. $\square$

Proposition 2

The BBGKY hierarchy with the collision term given by (29) is time-reversal invariant.

Proof

Recall that the BBGKY hierarchy has the form, for $k=1, \ldots N$:

$$\begin{aligned} \frac{\partial \rho _{k,t}^{(a)}(x_1,\ldots ,x_k)}{\partial t} - \mathcal{H}_k \rho _{k,t}^{(a)}(x_1, \ldots , x_k) = \left( { \mathcal C}^{(a)} _{k, k+1} \rho _{k+1, t}^{(a)}\right) (x_1, \ldots , x_k). \end{aligned}$$

(60)

In this equation, we deal with $k$ particles (the momentum of the $k+1$th particle appears in (29) only as an integration variable). If we reverse sign of the momenta $\vec {p}_1, \ldots \vec {p}_k$ and the sign of $t$, it is easy to see the the left hand side of (66) changes sign. But here, the right-hand side (29) clearly changes sign too when we change sign of all momenta $\vec {p}_1, \ldots , \vec {p}_{k+1}$, due to the fact that the integration over the antisymmetric factor $\left( \vec {\omega }_{i, k+1}\cdot \big (\vec {p}_{k+1} - \vec {p}_i\big ) \right)$ in the integrand is extended over the entire surface of theunit sphere. More explicitly, if we use the notation $\bar{x}_i = (\vec {q}_i,- \vec {p}_i)$ along $x_i = (\vec {q}_i, \vec {p}_i)$, the transformed version of the left-hand side of Eq. (60) is

$$\begin{aligned}&-\frac{\partial }{\partial t} \rho _{k,-t}^{(a)}(\bar{x}_1,\ldots ,\bar{x}_k)+ \mathcal{H}_k \rho _{k,-t}^{(a)}(\bar{x}_1, \ldots , \bar{x}_k)= -\frac{\partial }{\partial t}\tilde{\rho }_{k,t}^{(a)}({x}_1,\ldots ,{x}_k)\nonumber \\&\quad +\, \mathcal{H}_k \tilde{\rho }_{k,t}^{(a)}({x}_1, \ldots , {x}_k), \end{aligned}$$

(61)

while the right-hand side transforms into:

$$\begin{aligned}&\left( \mathcal{C}^{(a)}_{k, k+1} \rho _{k+1,-t}^{(a)}\right) (\bar{x}_1, \ldots \bar{x}_k) = Na^2 \sum _{i=1}^k \int _{{\mathbb {R}}^3} d\vec {p}_{k+1} \int _{S^2} d\vec {\omega }_{i,k+1}\nonumber \\&\quad \left( \vec {\omega }_{i, k+1}\cdot \big (\vec {p}_{k+1} + \vec {p}_i\big ) \right) \rho _{k+1,-t}^{(a)}(\bar{x}_1, \ldots , \bar{x}_k , \vec {q}_{i} + a \vec {\omega }_{i,k+1},\vec {p}_{k+1}). \end{aligned}$$

(62)

Hence, if we rewrite the integration variable $\vec {p}_{k+1}$ as $-\vec {p}_{k+1}$, we obtain from (62):

$$\begin{aligned}&\left( \mathcal{C}^{(a)}_{k, k+1} \rho _{k+1,-t}^{(a)}\right) (\bar{x}_1, \ldots , \bar{x}_k) = \nonumber \\&\quad - Na^2 \sum _{i=1}^k \int _{{\mathbb {R}}^3}\!\!\!d\vec {p}_{k+1} \int _{S^2}\!\!\! d\vec {\omega }_{i,k+1} \left( \vec {\omega }_{i, k+1}\cdot \big ( \vec {p}_{k+1} - \vec {p}_i\big ) \right) {\rho }_{k+1,-t}^{(a)}(\bar{x}_1, \ldots , \bar{x}_k , \vec {q}_{i}\nonumber \\&\quad +\, a \vec {\omega }_{i,k+1} ,-\vec {p}_{k+1}) \, = \nonumber \\&\quad - Na^2 \sum _{i=1}^k \int _{{\mathbb {R}}^3}\!\!\!d\vec {p}_{k+1} \int _{S^2}\!\!\! d\vec {\omega }_{i,k+1} \left( \vec {\omega }_{i, k+1}\cdot \big ( \vec {p}_{k+1} - \vec {p}_i\big ) \right) \tilde{\rho }_{k+1,-t}^{(a)}({x}_1, \ldots , {x}_k , \vec {q}_{i}\nonumber \\&\quad +\, a \vec {\omega }_{i,k+1} ,\vec {p}_{k+1}). \end{aligned}$$

(63)

Comparing this with (29), we conclude

$$\begin{aligned} \left( \mathcal{C}^{(a)}_{k, k+1} \rho _{k+1,-t}^{(a)}\right) (\bar{x}_1, \ldots \bar{x}_k) = - ({ \mathcal C}^{(a)} _{k, k+1}\tilde{ \rho }_{k+1, t}^{(a)})({x}_1, \ldots , {x}_k). \end{aligned}$$

(64)

Putting (61) and (64) together, we see that the time-reversed version $\tilde{\rho }^{(a)}_{k,t}$ of an arbitrary solution $\rho ^{(a)}_{k,t}$ of (60) obeys the equivalent equation

$$\begin{aligned} -\frac{\partial }{\partial t}\tilde{\rho }_{k,t}^{(a)} + \mathcal{H}_k \tilde{\rho }_{k,t}^{(a)} = - { \mathcal C}^{(a)} _{k, k+1}\tilde{ \rho }_{k+1, t}^{(a)}. \end{aligned}$$

(65)

This shows that if $\{ \rho _{k+1, t}^{(a)}\, , t \in {\mathbb {R}}\}$ solves Eq. (60), then $\{ \tilde{ \rho }_{k+1, t}^{(a)}\, , t \in {\mathbb {R}}\}$ is a solution of the same equation, so that we can conclude that (60) is time-reversal invariant. $\square$

Proposition 3

Let the continuity across collisions condition (53) hold. Then, the BBGKY hierarchy with the collision term expressed by (32) is equal to the BBGKY hierarchy with the collision term expressed by (52). Furthermore, it is time-reversal invariant.

Proof

Recall that, after adopting the incoming configuration for collision points, the BBGKY hierarchy takes the form

$$\begin{aligned} \frac{\partial \rho _{k,t}^{(a)}(x_1,\ldots ,x_k)}{\partial t} - \mathcal{H}_k \rho _{k,t}^{(a)}(x_1, \ldots , x_k) = \left( { \mathcal C}^{(a)} _{k, k+1} \rho _{k+1, t}^{(a)}\right) (x_1, \ldots , x_k), \end{aligned}$$

(66)

where the left-hand side is the same as in (60), but the collision operator is now expressed by (32). On the other hand, when adopting the outgoing configuration for collision points, the collision operator is expressed by (52).

We first show that the BBGKY hierarchy with the collision term expressed by (32) is equal to the BBGKY hierarchy with the collision term expressed by (52). Since the only apparent difference between these hierarchies lies in the collision term, one just needs to show that the operator (32) is the same as (52). Let us focus on the former. Notice that the phase point $(x_{1}, \ldots , x_{i-1},\vec {q}_{i} ,\vec {p}_{i}^{\ \prime }, x_{i+1}, \ldots x_k, \vec {q}_{i} - a \vec {\omega }_{i, k+1} , \vec {p}_{k+1}^{\ \prime })$ as well as the phase point $(x_{1}, \ldots , x_{i-1},\vec {q}_{i} ,\vec {p}_{i}, x_{i+1}, \ldots x_k, \vec {q}_{i} + a \vec {\omega }_{i, k+1} , \vec {p}_{k+1})$ are pre-collision coordinates in the hemisphere $\vec {\omega }_{i,k+1} \cdot (\vec {p}_{i} -\vec {p}_{k+1}) \geqslant 0$. Therefore, if the continuity at collisions condition holds, it follows that (32) can be written as

$$\begin{aligned}&\left( \mathcal{C}^{(a)}_{k, k+1} \rho ^{(a)}_{k+1,t} \right) (x_1, \ldots , x_k) \nonumber \\&\quad = Na^{2} \sum _{i=1}^{k}\int _{{\mathbb {R}}^3} \!d\vec {p}_{k+1} \int _{\vec {\omega }_{i,k+1} \cdot (\vec {p}_{i} -\vec {p}_{k+1}) \geqslant 0} \! \!\!d \vec {\omega }_{i,k+1}\, \, \, \vec {\omega }_{i,k+1}\cdot \left( \vec {p}_{i}- \vec {p}_{k+1} \right) \nonumber \\&\quad \times \big [ \rho _{k+1,t}^{(a)}(x_{1}, \ldots , x_{i-1},\vec {q}_{i} ,\vec {p}_{i}, x_{i+1}, \ldots x_k, \vec {q}_{i} - a \vec {\omega }_{i, k+1} , \vec {p}_{k+1})\nonumber \\&\quad - \rho _{k+1,t}^{(a)}(x_{1}, \ldots , x_{i-1}, \vec {q}_{i} , \vec {p}_{i}^{\ \prime }, x_{i+1}, \ldots x_k, \vec {q}_{i} + a \vec {\omega }_{i, k+1}, \vec {p}_{k+1}^{\ \prime }) \big ], \end{aligned}$$

(67)

which is just equal to the collision term (52).

Next, we demonstrate that the BBGKY hierarchy with the collision term expressed by (32) is time-reversal invariant. Here, we are again dealing with $k$ particles, but in this case both incoming and outgoing momenta appear in the same formula, just as in the Boltzmann equation. And just as in the Boltzmann equation, one ought to take the outgoing momenta variables here as (implicit) functions of the incoming momenta: $(\vec {p}_i^{\ \prime }, \vec {p}_{k+1}^{\ \prime }) = T_{\omega _{i,k+1}} (\vec {p}_i, \vec {p}_{k+1})$. We now apply a combination of the arguments we used above to judge the time-reversal invariance of the Boltzmann equation and the BBGKY hierarchy in the version (60): we replace $t$ by $- t$ and $(x_1 , \ldots , x_k)$ by $(\bar{x}_1 , \ldots , \bar{x}_k)$. Since the left-hand side is the same as in (60), we draw the same conclusion: this side transforms into (61). But we have to scrutinize the behaviour of the right-hand side in more detail. This side transforms to:

$$\begin{aligned}&\left( \mathcal{C}^{(a)}_{k, k+1} \rho ^{(a)}_{k+1,-t} \right) (\bar{x}_1, \ldots , \bar{x}_k) \nonumber \\&\quad = d\vec {p}_{k+1} d \vec {\omega }_{i,k+1}\, \, \vec {\omega }_{i,k+1}\cdot \left( \vec {p}_{i}+\vec {p}_{k+1} \right) \nonumber \\&\quad \times \big [ \rho _{k+1,-t}^{(a)}(\bar{x}_{1}, \ldots , \bar{x}_{i-1},\vec {q}_{i} ,\vec {p}_{i}^{\ \prime }, \bar{x}_{i+1}, \ldots \bar{x}_k, \vec {q}_{i} - a \vec {\omega }_{i, k+1}, \vec {p}_{k+1}^{\ \prime })\nonumber \\&\quad -\rho _{k+1,-t}^{(a)}(\bar{x}_{1}, \ldots , \bar{x}_{i-1}, \vec {q}_{i} , -\vec {p}_{i}, \bar{x}_{i+1}, \ldots \bar{x}_k, \vec {q}_{i} + a \vec {\omega }_{i, k+1}, -\vec {p}_{k+1}) \big ], \end{aligned}$$

(68)

where, as before, the variables $(\vec {p}_i^{\ \prime }, \vec {p}_{k+1}^{\ \prime })$ are defined as $(\vec {p}_i^{\ \prime }, \vec {p}_{k+1}^{\ \prime }) = T_{\omega _{i,k+1}} (-\vec {p}_i, \vec {p}_{k+1})$. Repeating a similar step of our first argument, we perform a conventional transformation on the integration variables $\vec {p}_{k+1} \longrightarrow -\vec {p}_{k+1}$ and $\vec {\omega }_{_i,k+1} \longrightarrow - \vec {\omega }_{i, k+1}$ and use that the primed momenta transform as $(\vec {p}_i^{\ \prime },\vec {p}_{k+1}^{\ \prime }) \longrightarrow T_{\omega _{i,k+1}} (-\vec {p}_i, -\vec {p}_{k+1}) = (-\vec {p}_i^{\ \prime },-\vec {p}_{k+1}^{\ \prime })$ to rewrite the integral (68) as

$$\begin{aligned}&\left( \mathcal{C}^{(a)}_{k, k+1} \rho ^{(a)}_{k+1,-t} \right) (\bar{x}_1, \ldots , \bar{x}_k)\nonumber \\&\quad = Na^{2} \sum _{i=1}^{k}\int _{{\mathbb {R}}^3} \!d\vec {p}_{k+1} \int _{\vec {\omega }_{i,k+1} \cdot (\vec {p}_{i} -\vec {p}_{k+1}) \geqslant 0} \! \!\!d \vec {\omega }_{i,k+1}\, \vec {\omega }_{i,k+1}\cdot \left( \vec {p}_{i}- \vec {p}_{k+1} \right) \nonumber \\&\quad \times \big [ \rho _{k+1,-t}^{(a)}(\bar{x}_{1}, \ldots , \bar{x}_{i-1},\vec {q}_{i} ,-\vec {p}_{i}^{\ \prime }, \bar{x}_{i+1}, \ldots \bar{x}_k, \vec {q}_{i} + a \vec {\omega }_{i, k+1} , -\vec {p}_{k+1}^{\ \prime }) \nonumber \\&\quad -\rho _{k+1,-t}^{(a)}(\bar{x}_{1}, \ldots , \bar{x}_{i-1}, \vec {q}_{i} , -\vec {p}_{i}, \bar{x}_{i+1}, \ldots \bar{x}_k, \vec {q}_{i} - a \vec {\omega }_{i, k+1}, -\vec {p}_{k+1}) \big ]. \end{aligned}$$

(69)

Now, in analogy to the previous case (cf. Eqn. (64)), we must show that

$$\begin{aligned} \left( \mathcal{C}^{(a)}_{k, k+1} {\rho }_{k+1,-t}^{(a)}\right) (\bar{x}_{1}, \ldots , \bar{x}_{k}) \left. \displaystyle { =} \right. - \left( \mathcal{C}^{(a)}_{k, k+1} \tilde{\rho }^{(a)}_{k+1,t}\right) ({x}_{1}, \ldots , {x}_{k}). \end{aligned}$$

(70)

For this purpose, notice that the phase point $(\bar{x}_{1}, \ldots , \bar{x}_{i-1},\vec {q}_{i} ,-\vec {p}_{i}^{\ \prime }, \bar{x}_{i+1}, \ldots \bar{x}_k, \vec {q}_{i} + a \vec {\omega }_{i, k+1} , -\vec {p}_{k+1}^{\ \prime })$ as well as the phase point $(\bar{x}_{1}, \ldots , \bar{x}_{i-1},\vec {q}_{i} ,-\vec {p}_{i}, \bar{x}_{i+1}, \ldots \bar{x}_k, \vec {q}_{i} + a \vec {\omega }_{i, k+1} , -\vec {p}_{k+1})$ are pre-collision coordinates in the hemisphere $\vec {\omega }_{i,k+1} \cdot (\vec {p}_{i} -\vec {p}_{k+1}) \geqslant 0$, and thus one can apply the continuity at collisions condition (53) to both such points. The collision term in (67) can thus be written as

$$\begin{aligned}&\left( \mathcal{C}^{(a)}_{k, k+1} \rho ^{(a)}_{k+1,-t} \right) (\bar{x}_1, \ldots , \bar{x}_k) \nonumber \\&\quad = Na^{2} \sum _{i=1}^{k}\int _{{\mathbb {R}}^3} \!d\vec {p}_{k+1} \int _{\vec {\omega }_{i,k+1} \cdot (\vec {p}_{i} -\vec {p}_{k+1}) \geqslant 0} \! \!\!d \vec {\omega }_{i,k+1}\, \, \, \vec {\omega }_{i,k+1}\cdot \left( \vec {p}_{i}- \vec {p}_{k+1} \right) \nonumber \\&\quad \times \big [ \rho _{k+1,-t}^{(a)}(\bar{x}_{1}, \ldots , \bar{x}_{i-1},\vec {q}_{i} ,-\vec {p}_{i}, \bar{x}_{i+1}, \ldots \bar{x}_k, \vec {q}_{i} + a \vec {\omega }_{i, k+1} , -\vec {p}_{k+1})\nonumber \\&\quad -\rho _{k+1,-t}^{(a)}(\bar{x}_{1}, \ldots , \bar{x}_{i-1}, \vec {q}_{i} , -\vec {p}_{i}^{\ \prime }, \bar{x}_{i+1}, \ldots \bar{x}_k, \vec {q}_{i} - a \vec {\omega }_{i, k+1}, -\vec {p}_{k+1}^{\ \prime }) \big ]. \end{aligned}$$

(71)

If one then performs the (cosmetic) transformation of the integration variable $\vec {\omega }_{i, k+1}$ into $- \vec {\omega }_{i, k+1}$, one obtains

$$\begin{aligned}&\left( \mathcal{C}^{(a)}_{k, k+1} \rho ^{(a)}_{k+1,-t} \right) (\bar{x}_1, \ldots , \bar{x}_k)\nonumber \\&\quad = Na^{2} \sum _{i=1}^{k}\int _{{\mathbb {R}}^3} \!d\vec {p}_{k+1} \int _{\vec {\omega }_{i,k+1} \cdot (\vec {p}_{i} -\vec {p}_{k+1}) \geqslant 0} \! \!\!d \vec {\omega }_{i,k+1}\, \vec {\omega }_{i,k+1}\cdot \left( \vec {p}_{i}- \vec {p}_{k+1} \right) \nonumber \\&\quad \times \big [ \rho _{k+1,-t}^{(a)}(\bar{x}_{1}, \ldots , \bar{x}_{i-1},\vec {q}_{i} ,-\vec {p}_{i}, \bar{x}_{i+1}, \ldots \bar{x}_k, \vec {q}_{i} - a \vec {\omega }_{i, k+1} , -\vec {p}_{k+1}) \nonumber \\&\quad -\rho _{k+1,-t}^{(a)}(\bar{x}_{1}, \ldots , \bar{x}_{i-1}, \vec {q}_{i} , -\vec {p}_{i}^{\ \prime }, \bar{x}_{i+1}, \ldots \bar{x}_k, \vec {q}_{i} + a \vec {\omega }_{i, k+1}, -\vec {p}_{k+1}^{\ \prime }) \big ], \end{aligned}$$

(72)

which is equal to $- \left( \mathcal{C}^{(a)}_{k, k+1} \rho ^{(a)}_{k+1,-t} \right) (\bar{x}_1, \ldots , \bar{x}_k)$, as one can see by contrasting the above equation with (67). By recalling the definition of $\tilde{\rho }_{k, t}^{(a)}$, we conclude that

$$\begin{aligned} \left( \mathcal{C}^{(a)}_{k, k+1} \rho ^{(a)}_{k+1,-t} \right) (\bar{x}_1, \ldots , \bar{x}_k)&= - \left( \mathcal{C}^{(a)}_{k, k+1} \tilde{\rho }^{(a)}_{k+1,t} \right) (x_1, \ldots , x_k). \end{aligned}$$

(73)

It has thus been shown that the BBGKY hierarchy with the collision term expressed by (32) is time-reversal invariant. Equivalently, the BBGKY hierarchy with the collision term expressed by (52) is time-reversal invariant too. $\square$