reperiendi

Imaginary Time 2

Posted in Chemistry, General physics, Math by Mike Stay on 2010 July 26

Another part of the analogy I started here, but this time using inverse temperature instead of imaginary time. It describes a thermometer where a mass changes position with temperature. I’m guessing this stuff only applies when the temperature is changing adiabatically.

Thermometer (unitless temperature):
\displaystyle \beta [1] inverse temperature (unitless)
\displaystyle y(\beta) [m] y coordinate
\displaystyle r [kg/s^2 K] spring constant * temp unit conversion
\displaystyle v(\beta) = \frac{dy(\beta)}{d\beta} [m] how position changes with (inverse) temperature
\displaystyle F(\beta) = r \; v(\beta) [kg m/s^2 K] force per Kelvin
\displaystyle T(\beta) = \frac{r}{2}v(\beta)^2 [kg m^2/s^2 K] stretching energy per Kelvin
\displaystyle V(\beta) [kg m^2/s^2 K] potential energy per Kelvin
\displaystyle S = \int (T + V)(\beta) \; d\beta

[kg m^2/s^2 K] entropy
Thermometer:
\displaystyle \beta [1/K] inverse temperature
\displaystyle y(\beta) [m] y coordinate
\displaystyle r [kg/s^2 K^2 = bits/m^2 K] how information density changes with temp
\displaystyle v(\beta) = \frac{dy(\beta)}{d\beta} [m K] how position changes with (inverse) temperature
\displaystyle F(\beta) = r \; v(\beta) [kg m/s^2 K = bits/m] force per Kelvin
\displaystyle T(\beta) = \frac{r}{2}v(\beta)^2 [kg m^2/s^2 = bits K] stretching energy = change in stretching information with invtemp
\displaystyle V(\beta) [kg m^2/s^2 = bits K] potential energy = change in potential information with invtemp
\displaystyle S = \int (T + V)(\beta) \; d\beta

[bits] entropy

I assume that the dynamics of such a system would follow a path where \displaystyle \delta S=0; is that a minimum-entropy path or a maximum?

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