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# Dungeon Madness By Dakota Krout (.ePUB) Pdf Fixed

Dungeon Madness By Dakota Krout (.ePUB) Pdf Fixed Dungeon Madness By Dakota Krout (.ePUB) Pdf

Size: 5937 KB Type: PDF, ePub, eBook.. Dungeon Madness by Dakota Krout (.ePUB) pdf.}\,, \label{eq:dmu}$$where f(\vec{v}) and g(\vec{v}) are the distribution functions of {\mathcal V} in velocity space and \mu is the mean molecular weight of a cell. Note that the net rate of momentum loss by viscous damping is given by$$\frac{d\langle\vec{v}\rangle}{dt}=-\frac{1}{\tau}\vec{v}=-\frac{\mu}{\tau}\vec{U}\,, \label{eq:dv}$$which, in conjunction with Eq. ($eq:dv1$), gives$$\frac{d \vec{U}}{dt}=-\vec{U}\frac{1}{\tau}-\frac{d\langle\vec{v}\rangle}{dt}= \vec{U}\left(-\frac{1}{\tau}-\frac{\mu}{\tau}\right)\,. \label{eq:dv2}$$As a consequence,$$\frac{d\langle\vec{v}\rangle}{dt}=\left\langle\frac{d\langle\vec{v}\rangle}{dt}\right\rangle= \frac{d \vec{U}}{dt}\,. \label{eq:noise2}$$Equation ($eq:noise2$) can be recast as a random force,$$\frac{d\vec{v}}{dt}=\vec{U}+\vec{f}_{\rm r}\,, \label{eq:dvdt}$$where \vec{f}_{\rm r} is a random variable drawn from the Gaussian distribution with zero mean and$$\overline{f_i({\mathcal V})}=0\,, \qquad \overline{f_i({\mathcal V})f_j({\mathcal V’})}=\frac{\mu}{\tau}\delta_{ij}\delta({\mathcal V}-{\mathcal V’})\,, \label{eq:therm} where
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