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

The variables dndmu, dndT, and dsdT correspond to

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

respectively. All other derivatives can be expressed simply in terms of these three. Note that volume derivatives are trivial, since both the entropy and number scale linearly with the volume.


Derivatives wrt to chemical potential and temperature:

There is a Maxwell relation

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

The pressure derivatives are trivial

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

The energy density derivatives are related through the thermodynamic identity:

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

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


Other derivatives:

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

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

As noted in Particles in the User's Guide for O2scl_part , we work in units so that $ \hbar = c = k_B = 1 $. In this case, $ c_V $ is unitless as defined here. To compute $ c_V $ 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

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

We can then construct a function

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

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.

For, $ c_P $, defined as

\[ c_P = \frac{T}{N} \left( \frac{\partial S}{\partial T} \right)_{N,P} \]

(which is also unitless) we can write functions

\[ S=S(N,T,V) \qquad \mathrm{and} \qquad V=V(N,P,T) \]

which imply

\[ \left( \frac{\partial S}{\partial T} \right)_{N,P} = \left( \frac{\partial S}{\partial T} \right)_{N,V} + \left( \frac{\partial S}{\partial V} \right)_{N,T} \left( \frac{\partial V}{\partial T} \right)_{N,P} \, . \]

Thus we require the derivatives

\[ \left( \frac{\partial S}{\partial T} \right)_{N,V} , \left( \frac{\partial S}{\partial V} \right)_{N,T} , \qquad\mathrm{and}\qquad \left( \frac{\partial V}{\partial T} \right)_{N,P} \, . \]

To compute the new entropy derivatives, we can write

\[ S=S(\mu(N,T,V),T,V) \]

to get

\[ \left( \frac{\partial S}{\partial T} \right)_{N,V} = \left( \frac{\partial S}{\partial \mu} \right)_{T,V} \left( \frac{\partial \mu}{\partial T} \right)_{N,V} + \left( \frac{\partial S}{\partial T} \right)_{\mu,V} \, , \]

and

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

These require the chemical potential derivatives which have associated Maxwell relations

\[ \left( \frac{\partial \mu}{\partial T} \right)_{N,V} = -\left( \frac{\partial S}{\partial N} \right)_{T,V} \qquad\mathrm{and}\qquad \left( \frac{\partial \mu}{\partial V} \right)_{N,T} = -\left( \frac{\partial P}{\partial N} \right)_{T,V} \, . \]

Finally, we can rewrite the derivatives on the right hand sides in terms of derivatives of functions of $ \mu, V $ and $ T $,

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

and

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

The volume derivative,

\[ \left( \frac{\partial V}{\partial T} \right)_{N,P} \, , \]

is related to the coefficient of thermal expansion, sometimes called $ \alpha $,

\[ \alpha \equiv \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_{N,P} \, . \]

We can rewrite the derivative

\[ \left( \frac{\partial V}{\partial T} \right)_{N,P} = -\left( \frac{\partial P}{\partial T} \right)_{N,V} \left( \frac{\partial P}{\partial V} \right)_{N,T}^{-1} \, . \]

The first term can be computed from the Maxwell relation

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

where the entropy derivative was computed above. The second term (related to the inverse of the isothermal compressibility, $ \kappa_T \equiv (-1/V) (\partial V/\partial P)_{T,N} $ can be computed from the function $ P = P(\mu(N,V,T),V,T) $

\[ \left( \frac{\partial P}{\partial V} \right)_{N,T} = \left( \frac{\partial P}{\partial \mu} \right)_{T,V} \left( \frac{\partial \mu}{\partial V} \right)_{N,T} + \left( \frac{\partial P}{\partial V} \right)_{\mu,T} \]

where the chemical potential derivative was computed above.

The results above can be collected to give

\[ \left( \frac{\partial S}{\partial T} \right)_{N,P} = \left( \frac{\partial S}{\partial T} \right)_{\mu,V} + \frac{S^2}{N^2} \left( \frac{\partial N}{\partial \mu} \right)_{T,V} - \frac{2 S}{N} \left( \frac{\partial N}{\partial T} \right)_{\mu,V} \, , \]

which implies

\[ c_P = \frac{T}{n} \left( \frac{\partial s}{\partial T} \right)_{\mu,V} + \frac{s^2 T}{n^3} \left( \frac{\partial n}{\partial \mu} \right)_{T,V} - \frac{2 s T}{n^2} \left( \frac{\partial n}{\partial T} \right)_{\mu,V} \, , \]

This derivation also gives the well-known relationship between the specific heats at constant volume and constant pressure,

\[ c_P = c_V + \frac{T \alpha^2}{n \kappa_T} \, . \]

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 260 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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