deriv_part Class Reference

Storage for deriviatives wrt $ \mu $ and T. More...

#include <deriv_part.h>

Inheritance diagram for deriv_part:

sn_classical sn_fermion sn_nr_fermion

Detailed Description

Storage for deriviatives wrt $ \mu $ and T.

The variables dndmu, dndT, and dsdT correspond to

\[ \left(\frac{d n}{d \mu}\right)_{T}, \quad \left(\frac{d n}{d T}\right)_{\mu}, \quad \mathrm{and} \quad \left(\frac{d s}{d T}\right)_{\mu} \]

respectively.

All other derivatives can be expressed simply in terms of these three.


Derivatives wrt to chemical potential and temperature:

There is a Maxwell relation

\[ \left(\frac{d s}{d \mu}\right)_T = \left(\frac{d n}{d T}\right)_{\mu} \]

The pressure derivatives are trivial

\[ \left(\frac{d P}{d \mu}\right)_{T}=n, \quad \left(\frac{d P}{d T}\right)_{\mu}=s \]

The energy density derivatives are related through the thermodynamic identity:

\[ \left(\frac{d \varepsilon}{d \mu}\right)_{T}= \mu \left(\frac{d n}{d \mu}\right)_{T}+ T \left(\frac{d s}{d \mu}\right)_{T} \]

\[ \left(\frac{d \varepsilon}{d T}\right)_{\mu}= \mu \left(\frac{d n}{d T}\right)_{\mu}+ T \left(\frac{d s}{d T}\right)_{\mu} \]


Other derivatives:

Note that the derivative of the entropy with respect to the temperature above is not the specific heat, $ c_V $. The specific heat is

\[ C_V = \frac{T}{N} \left( \frac{\partial S}{\partial T} \right)_{V,N} = \frac{T}{n} \left( \frac{\partial s}{\partial T} \right)_{V,n} \]

To compute the specific heat in terms of the derivatives above, note that the descendants of deriv_part provide all of the thermodynamic functions in terms of $ \mu, V $ and $ T $, so we have

\[ s=s(\mu,V,T) \quad \mathrm{and} \quad n=n(\mu,V,T) \, . \]

We can then construct a function

\[ s=s[\mu(n,V,T),V,T] \]

and then write the required derivative directly

\[ \left(\frac{\partial s}{\partial T}\right)_{n,V} = \left(\frac{\partial s}{\partial \mu}\right)_{T,V} \left(\frac{\partial \mu}{\partial T}\right)_{n,V} + \left(\frac{\partial s}{\partial T}\right)_{\mu,V} \, . \]

Now we use the identity

\[ \left(\frac{\partial \mu}{\partial T}\right)_{n,V} = - \left(\frac{\partial n}{\partial T}\right)_{\mu,V} \left(\frac{\partial n}{\partial \mu}\right)_{T,V}^{-1} \, , \]

and the Maxwell relation above to give

\[ C_V = \frac{T}{n} \left[ \left(\frac{\partial s}{\partial T}\right)_{\mu,V} -\left(\frac{\partial n}{\partial T}\right)_{\mu,V}^2 \left(\frac{\partial n}{\partial \mu}\right)_{T,V}^{-1} \right] \]

which expresses the specific heat in terms of the three derivatives which are given.

Note that this is the specific heat per particle, and has no units. If specific heat per unit volume is required, you must multiply by the number density.

No derivative with respect to the bare mass is given, since classes cannot know how to relate the effective mass to the bare mass.

Definition at line 136 of file deriv_part.h.


Data Fields

double dndmu
 Derivative of number density with respect to chemical potential.
double dndT
 Derivative of number density with respect to temperature.
double dsdT
 Derivative of entropy density with respect to temperature.
double dndm
 Derivative of number density with respect to the effective mass.

The documentation for this class was generated from the following file:

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