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#### Peter Leopold

##### Guest

Peter Leopold Asks:

This question appeared in a study guide for my graduate level written exam in physics. (It may have been the one from the University of Chicago.) I see that a similar question was asked here What is the average value of time since last collision in Drude model? but the discussion did not focus on the aspect that I'm going to present.

A particle in a gas undergoes random collisions with other gas particles. The inter-collision times are described by an exponential random process. Consider a particle at time $t_0$. The distribution of times until the next collision is exponentially distributed $f(t | \tau) = (1/\tau)\exp(-(t-t_0)/\tau)$. Clearly, $\mathbb{E}[t-t_0] = \tau$. But by time reversal symmetry, this also describes the time since the last collision. So which is it? Is the mean time between collisions $\tau$ or $2\tau$?

The question always felt like a swindle $-$ in part because I was unable to track down a generally agreed upon answer. (This was long before stackoverflow existed.)

My take is that

The problem is with the exact definition of the term "mean inter-collision time." As it can be defined to be 1) unconditionally "time to next collision", then it is $\tau$ (but it makes no reference to the last collision, so perhaps this definition is non-responsive to the name) or 2) conditional on a collision having just occurred, "time to next collision" = $\tau$, or 3) unconditionally, "time since last collision to next collision," for a randomly selected particle whose past is as unknown as its future, in which case it is in fact $2\tau.$

Is there a consensus on this point here?

*Conditional and unconditional interpretations of mean inter-collision times*This question appeared in a study guide for my graduate level written exam in physics. (It may have been the one from the University of Chicago.) I see that a similar question was asked here What is the average value of time since last collision in Drude model? but the discussion did not focus on the aspect that I'm going to present.

A particle in a gas undergoes random collisions with other gas particles. The inter-collision times are described by an exponential random process. Consider a particle at time $t_0$. The distribution of times until the next collision is exponentially distributed $f(t | \tau) = (1/\tau)\exp(-(t-t_0)/\tau)$. Clearly, $\mathbb{E}[t-t_0] = \tau$. But by time reversal symmetry, this also describes the time since the last collision. So which is it? Is the mean time between collisions $\tau$ or $2\tau$?

The question always felt like a swindle $-$ in part because I was unable to track down a generally agreed upon answer. (This was long before stackoverflow existed.)

My take is that

*conditional on knowing that the particle just endured a collision*, the mean time to the next collision is indeed $\tau.$ But unconditionally, the time since/until the last/next collision is properly $\tau$, so we compute the last-to-next time to be $2\tau.$The problem is with the exact definition of the term "mean inter-collision time." As it can be defined to be 1) unconditionally "time to next collision", then it is $\tau$ (but it makes no reference to the last collision, so perhaps this definition is non-responsive to the name) or 2) conditional on a collision having just occurred, "time to next collision" = $\tau$, or 3) unconditionally, "time since last collision to next collision," for a randomly selected particle whose past is as unknown as its future, in which case it is in fact $2\tau.$

Is there a consensus on this point here?

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