Let's say I have the $i$'th gas molecule with velocity $\vec v_i(t)$ at time $t$. To find the net displacement $s_i$ we integrate with respect to $t$:

$$ \vec s_i = \int_{0}^{t} \vec v_i(t') dt' $$

Since, the gas molecule under goes collisions. Initially the $i$'th gas molecule has velocity $v_{i,1}$ for time interval $t_1$ and then velocity $v_{i,2}$ for time interval $t_2$ and so on ...

$$ \int_{0}^{t} \vec v_i(t') dt' = \vec v_{i,1} t_1 + \vec v_{i,2} t_2 + \dots + \vec v_{i,n} t_n $$

Now, we can write the velocity as the infinitesimal ratio of displacement and time:

$$ \int_{0}^{t} \vec v_i(t') dt' = t_1 \frac{d \vec x_{i,1}}{dt} + t_2 \frac{d \vec x_{i,1}}{dt} + \dots + t_n \frac{dx_{i,n}}{dt} $$

Multiplying both sides by $dt$:

$$ \int_{0}^{t} \Big( \int_{0}^{t''} \vec v_i(t') dt' \Big) dt'' = \int^{ \vec x_i(t)}_{\vec x_i(0)} t_1 d \vec x_{i,1} + t_2 d \vec x_{i,2} + \dots + t_n dx_{i,n} $$

Using some number theory and analysis and assuming the $\vec v_i$ which appear follow a particular distribution:

$$ \int_{0}^{t} \Big( \int_{0}^{t''} \vec v_i(t') dt' \Big) dt'' = \langle t_c \rangle \int^{ \vec x_i(t)}_{\vec x_i(0)} d \vec x_i $$

where $\langle t_c \rangle$ is the average time interval for a collision. Since we have only done this for one particle we now proceed to do this over all the particles in a volume $V$ (and assume the collision rate in the block of volume is the same/the temperature is the same in the volume $V$).

$$ \int_{0}^{t} \Big( \int_{0}^{t''} \sum_{i=\text{all particle in $V$}}m_i \vec v_i(t') dt' \Big) dt'' /M = \langle t_c \rangle \sum_{i=\text{all particle in $V$}} \int_{ \vec x_i(0)}^{\vec x_i(t)} m_i d \vec x_i /M $$

where $M$ is the total mass $M$ in $V$ and $m_i$ is the mass of the $i$'th particle. Since the center of mass is immune to collisions:

$$ \int_{0}^{t} \Big( \int_{0}^{t''} \vec v_{CM}(t') dt' \Big) dt'' = \langle t_c \rangle \int_{0}^t \vec v_{CM} (t'') dt'' $$

where $\vec v_{CM}$ is the velocity of the center of mass and we used:

$$ \sum_{i=\text{all particle in $V$}} m_i \int_{ \vec x_i(0)}^{\vec x_i(t)} d \vec x_i /M = \sum_i m_i \int_0^t ( \vec v_{i,1} + \vec v_{i,2} + \vec v_{i,3} + \dots ) dt /M $$


Is the derivation correct? Is there some clever computational experiment way to verify this?

  • $\begingroup$ This discussion has been moved to chat $\endgroup$
    – Tyberius
    Jul 7 at 13:25

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