A particle data class with derivatives. More...
#include <part_deriv.h>
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_deriv & | operator= (const part_deriv &p) |
Copy construction with operator=() | |
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part (const part &p) | |
Copy constructor. | |
part & | operator= (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. | |
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double | g |
Degeneracy (e.g. spin and color if applicable) | |
double | m |
Mass. | |
double | n |
Number density. | |
double | ed |
Energy density. | |
double | pr |
Pressure. | |
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) | |
This class adds the derivatives dndmu, dndT, and dsdT, which correspond to
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
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
The pressure derivatives are trivial
The energy density derivatives are related through the thermodynamic identity:
Other derivatives:
Note that the derivative of the entropy with respect to the temperature above is not the specific heat per particle, . The specific heat per particle is
As noted in Particles in the User's Guide for O2scl_part , we work in units so that . In this case,
is unitless as defined here. To compute
in terms of the derivatives above, note that the descendants of part_deriv provide all of the thermodynamic functions in terms of
and
, so we have
We can then construct a function
and then write the required derivative directly
Now we use the identity
and the Maxwell relation above to give
which expresses the specific heat in terms of the three derivatives which are given.
For, , defined as
(which is also unitless) we can write functions
which imply
Thus we require the derivatives
To compute the new entropy derivatives, we can write
to get
and
These require the chemical potential derivatives which have associated Maxwell relations
Finally, we can rewrite the derivatives on the right hand sides in terms of derivatives of functions of and
,
and
The volume derivative,
is related to the coefficient of thermal expansion, sometimes called ,
We can rewrite the derivative
The first term can be computed from the Maxwell relation
where the entropy derivative was computed above. The second term (related to the inverse of the isothermal compressibility, can be computed from the function
where the chemical potential derivative was computed above.
The results above can be collected to give
which implies
This derivation also gives the well-known relationship between the specific heats at constant volume and constant pressure,
In the case where the particle is interacting, the derivative of the density with respect to the effective mass is
This relation holds whether or not the mass is included in the chemical potential , 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.
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