Public Member Functions | Public Attributes | List of all members
o2scl::part_deriv Class Reference

A particle data class with derivatives. More...

#include <part_deriv.h>

Inheritance diagram for o2scl::part_deriv:
o2scl::part o2scl::fermion_deriv

Public Member Functions

 part_deriv (double mass=0.0, double dof=0.0)
 Make a particle of mass mass and degeneracy dof.
 part_deriv (const part_deriv &p)
 Copy constructor.
part_derivoperator= (const part_deriv &p)
 Copy construction with operator=()
- Public Member Functions inherited from o2scl::part
 part (const part &p)
 Copy constructor.
partoperator= (const part &p)
 Copy construction with operator=()
 part (double mass=0.0, double dof=0.0)
 Make a particle of mass mass and degeneracy dof.
virtual void init (double mass, double dof)
 Set the mass mass and degeneracy dof.
virtual void anti (part &ap)
 Make ap an anti-particle with the same mass and degeneracy. More...
virtual const char * type ()
 Return string denoting type ("part")

Public Attributes

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.
- Public Attributes inherited from o2scl::part
double g
 Degeneracy (e.g. spin and color if applicable)
double m
double n
 Number density.
double ed
 Energy density.
double pr
double mu
 Chemical potential.
double en
 Entropy density.
double ms
 Effective mass (Dirac unless otherwise specified)
double nu
 Effective chemical potential.
bool inc_rest_mass
 If true, include the mass in the energy density and chemical potential (default true)
bool non_interacting
 True if the particle is non-interacting (default true)

Detailed Description

This class adds the derivatives dndmu, dndT, and dsdT, which 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 first-order thermodynamic derivatives can be expressed in terms of the first three derivatives. In the case that the particle is interacting (i.e. part::non_interacting is false), then the derivatives which are computed are

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

If the particles are interacting, no derivative with respect to the bare mass is given, since classes cannot know how to relate the effective mass to the bare mass.

Other derivatives with respect 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 part_deriv 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

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


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

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 $,

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


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

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,

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

In the case where the particle is interacting, the derivative of the density with respect to the effective mass is

\[ \left(\frac{dn}{dm^{*}}\right)_{\mu,T} = \left(\frac{3 n}{m^{*}}\right) - \frac{T}{m^{*}} \left(\frac{dn}{dT}\right)_{m^{*},\mu} - \frac{\nu}{m^{*}} \left(\frac{dn}{d\mu}\right)_{m^{*},T} \]

This relation holds whether or not the mass is included in the chemical potential $ \nu $, as the rest mass is held constant even though the effective mass is varying. This relation also holds in the case where the particle is non-interacting, so long as one does not allow the rest mass in the chemical potential to vary. This derivative is useful, for example, in models of quark matter where the quark mass is dynamically generated.

Definition at line 285 of file part_deriv.h.

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

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