# 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): $\displaystyle \beta$  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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