Suppose we make many very small clocks such that they're subject to thermal motion.
To make things simple suppose at some moment $t_0$ all of their times are synced (yes they're in close proximity). We then check on them at some later time and graph the distribution of the various time measurements.
It seems evident that a clock not moving at all (in our reference frame) will record the greatest time change, and there will be a distribution of others that record various lesser time intervals. Clearly such a distribution will be dependent upon the mass and temperature of the clocks, but I was curious about the general shape of such a distribution. Any thoughts or places I could look?
This is a classical system, I get that quantum theory would generally come into play here
EDIT:
This should be solvable utilizing the Maxwell-Boltzmann distribution:
$$f(v)=\left(1/2\pi a\right)^{3}4\pi v^{2}e^{\frac{-1}{2}\left(\frac{v}{a}\right)^{2}}$$
where $a=\sqrt{kT/m}$ (T and m being the temperature and mass of our clocks respectively). The time measured by a randomly chosen clock is then:
$$\Delta t=\int_{t_{2}}^{t_{1}}dt\gamma^{-1}$$
Where dt is the observers' (non moving clocks') proper time and the Lorentz factor gamma is going to depend upon the probablity of the particle having a particular speed at some particular time. I'm not sure how to proceed with this, but already it seems like an interesting problem if one considers that our consituent parts are continually getting “smeared” out in time. Surely there's a physically measureable consequence of this. Or maybe I should use the random walk?
EDIT 1
To address Rennie's comments below, it is claimed that the clocks will go toward the same time. ie. all of their individual time-average velocities will converge. The issue with this is that for Any arbitrarily large (but non-infinite) random walk in velocity space, the average need not fall on zero, and in fact there's always a finite probability of it falling quite far from zero (the more steps in the walk the further it could possibly be, but yes that probablity also goes down).
Furthermore, if the clocks were in equilibrium prior to syncing their times (a reasonable proposition), the origin of each clocks random walk (in velocity space) would vary by a probablity depending upon the initial (Maxwell-Boltzmann) velocity distribution, such that even after an infinite amount of time the clocks would be very out of sync.
I'm asking about the shape of such a distribution for a purely thermal system as it pertains to the time interval experienced by such a body relative to another having maintained an inertial frame consistently. What is the distribution of the clocks measurements in time. I have trouble believing they would all be synced as Rennie says, such behavior goes against a maximal entropy state