Annus MirabilisEinstein in 1905
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brownian motion

The equation explained

∂pν∂t=φν(p1…pl)(ν=1,2…l)\frac{\partial p_\nu}{\partial t} = \varphi_\nu(p_1 \ldots p_l) \quad (\nu = 1, 2 \ldots l)
In wordsHow fast one of the variables that fix the state of a system changes with the time is given by a function of the present values of all those variables.OverviewSection 2 begins with the kind of system its argument will use: one whose state is fixed by a list of variables, whose equations of change give each variable's rate as a function of the present values of them all, and whose flow in the space of those variables neither crowds together nor spreads out.Full explanationThe section works in the framework of Einstein's two earlier papers on the foundations of thermodynamics, which the footnote cites and which, it says, a reader needs neither for this section nor for the results of the paper. Let p1p2…plp_1 p_2 \ldots p_l be state variables of a physical system which completely determine its momentary state, for example the coordinates and velocity components of all the atoms of the system. The complete system of the equations of change of those variables is taken to have the form ∂pν∂t=φν(p1…pl)\frac{\partial p_\nu}{\partial t} = \varphi_\nu(p_1 \ldots p_l), one equation for each ν\nu from 1 to ll, and the same sentence adds the condition ∑∂φν∂pν=0\sum \frac{\partial \varphi_\nu}{\partial p_\nu} = 0. The first form says that where the system goes next is settled by where it is now, and not by the time or by its history. The second says that the flow in the space of the state variables has no sources and no sinks. Given both, the text says, the entropy of the system is the expression of the next display.Every step∂pν∂t=φν(p1…pl)(ν=1,2…l)\frac{\partial p_\nu}{\partial t} = \varphi_\nu(p_1 \ldots p_l) \quad (\nu = 1, 2 \ldots l)As printed. There are ll equations, one for each state variable, and the right side of each holds only the present values p1…plp_1 \ldots p_l.∑∂φν∂pν=0\sum \frac{\partial \varphi_\nu}{\partial p_\nu} = 0The condition printed in the same sentence, summed over the ll variables. Each term measures how much the flow along one variable's direction stretches as it moves; their sum vanishing means a set of states carried by the flow keeps its size.Historian's marginThe footnote to the section heading cites Einstein's own papers, Ann. d. Phys. 9, p. 417 (1902) and 11, p. 170 (1903), and says that for understanding the results of the present work, knowledge of those papers and of this section is dispensable. The section is the one place in the paper where the osmotic pressure of the first section is derived rather than assumed.Also

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