Re: [Phys-L] ?conservation of _internal_ energy

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding the issue JD brought up:

> Maybe you folks can help me out here.
>
> Most (albeit not all) thermo books emphasize the «internal energy»,
> denoted U or E_in.  It equals the plain old energy E minus the
> center-of-mass kinetic energy of the parcel minus potential energy
> terms such as gravity.
>
> I have never been able to make sense of this «internal energy».

The way I understand the situation is what is typically/usually called the 'internal energy' in thermodynamics/statistical physics is simply the total (plain old as JD says) energy but *as seen* in a particular frame in which the system's center-of-mass is at rest and not rotating, or, more specifically, the  frame is one in which the system's total momentum is zero and the total angular momentum about its CM is zero.  Such a frame may be either inertial or non-inertial depending on whether the system as a whole is undergoing any accelerated or rotational motions.  The internal energy in such a frame may well contain potential energy terms from couplings of the system's degrees of freedom with any external frame-induced force fields such as gravitational and centrifugal fields.  I think the reason for doing this tends to be mostly for convenience of analysis, and that convenience is an artifact of the Poincare invariance of nature (in situations of negligible space-time curvature).  An explanation follows.

An object (with positive mass) that is isolated/disconnected from its external environment is at rest in some inertial frame, and in any inertial frame it possesses a number of extensive constants of motion stemming from a combination of Noether's Theorem and Poincare invariance.  Since the Poincare group has 10 independent continuous parameters there are 10 of these constants of motion.  Invariance under space-time displacements causes the energy-momentum 4 vector to have 4 separately conserved components (E, p_x, p_y, p_z).  Also invariance under Lorentz transformations produces 6 more conserved quantities.  These 6 can be further separated into the 3 components of angular momentum from pure rotations in space, and 3 more components from pure boosts.

When we do equilibrium stat mech on an isolated system, (i.e. one described by a microcanonical ensemble) we need to count the system's microstates that are consistent/compatible with its macrostate.  If a system's macrostate has some conserved extensive quantities that can't change under internal rearrangements and interactions among its microstates then we are restricted to only count those microstates in the system's phase space (here the term 'phase space' is taken loosely to also/possibly include the distinct orthogonal dimensions in Hilbert space as its elements if the system's microstates need to be described by quantum mechanics) that lie on an appropriate restricted 'surface' or subspace defined by the constant values of those macroscopic constants.

If the system is not fully isolated but is in weak interaction with an environment of a number of fixed intensive potentials, (i.e. temperature, pressure, chemical potential, etc.) then the restriction on the subspace of the full phase space for which the counting is done is relaxed/removed for those extensive conserved parameters which are conjugate to the appropriate fixed environmental intensive parameters.  For instance if the weak interactions with the environment can exchange energy with an environment whose temperature is fixed then the microstate counting process is *not* restricted to the microstates with a given fixed extensive energy, but inculdes all microstates of any extensive energy allowable by the Hamiltonian.  But along with this relaxation of constraint the counting process doesn't just include a bare enumeration of allowed microstates, but instead counts Boltzmann factor exponentials whose exponent is -E/(kT) where E is the microstate's extensive energy, k is Boltzmann's constant, and T is the fixed temperature of the environment.  The resulting sum is the canonical partition function rather than the total number of allowed microstates in the microcanonical ensemble of the isolated system.  Likewise, if the system is allowed to exchange both energy and particles with an environment of fixed temperature and fixed chemical potential then both the constant energy and constant particle number constraints on the enumeration are relaxed and all energies and all particle numbers are included in the enumeration, and the exponent of the Boltzmann factors for the enumerated microstates is -(E - [mu]N)/(kT) where [mu] is the environment's chemical potential and N is the extensive number of particles in each enumerated microstate.  In the latter case the sum of Boltzmann factors over all the microstates is the grand canonical partition function.

So if one works in the microcanonical ensemble of isolated systems and the members in the ensemble all are moving through space with a given nonzero total momentum vector and are rotating with a given nonzero angular momentum bivector then the enumeration of microstates in phase space is restricted to a subspace of a particular fixed total value for each of the extensive parameters (E, p_x, p_y, p_z, L_x, L_y, L_z).  Such a multi-dimensional set of restrictions on the relevant phase space can be a real nuisance when doing the microstate enumeration.  Likewise, if one works in some other ensemble non-isolated ensemble exchanging some conserved extensive quantities with the environment having some corresponding fixed conjugate intensive potentials the Boltzmann factor exponents need to have terms for all such exchanged quantities multiplied by some fixed intennsive quantity and divided by k. If the environment is allowed to exchange momentum and/or angular momentum with the system then the exponents of the summed Boltzmann factors needs terms involving each of those exchanged momentum and angular momentum components. Evaluating the sum of such macroscopic functions of the microstate in the exponents and adding the Boltzmann factors evaluated all up can also be a real nuisance.  Thus switching to a different ensemble may not help much.

Now if we do our stat mech on a system as seen in a reference frame that is non-rotating and at rest then the system's total extensive momentum and angular momentum components are all zero in that frame. One way to keep a system at rest is to confine it in a container with reflective walls and is not moving.  Often doing the thermo on a system in a fixed container is a lot easier than dealing with its CM motion besides dealing with its fixed energy and particle number.  Another way to keep a system at rest for which the use of a container is not appropriate (e.g. an expanding gas cloud) is to rewrite the system's Hamiltonian in a coordinate system that treats the CM degrees of freedom as their own dynamical variables, and treat the rest of the degrees of freedom in terms of relative coordinates measured WRT the CM.  Then just do the thermo problem on the relative motion degrees of freedom & treat the CM degrees of freedom as being at rest with the location of the CM at the coordinate origin.  If we can solve the thermo problem for such a rest system and if our actual system is translating at a nonzero velocity in some direction then the relatively simple kinematics of that overall CM motion can be appended to the internal themo behavior to get a complete description of the system's behavior.  We just perform a boost to the appropriate frame in which the system is actually translating.  If our system was actually rotating then the corresponding rest system would need to have the appropriate centrifugal field included in the rest system's Hamiltonian before the thermo/stat mech is done.  And if the system is being accelerated as a whole (or in a static gravity field of a planet) then the appropriate inertial force fields also would need to be included in the rest system's Hamiltonian before the thermo problem is analysed.  Boosting and rotating a motion is fairly straightforward compared to messy stat mech enumerations and summations, and such boosting and/or rotating can be done if necessary after the messy internal thermal phyiscs is dealt with.

In short, confining one's attention to a reference frame in which the system is at rest and not rotating simplifies the corresponding thermo problem a lot over what it would be if the CM was translating or the system was rotating with all those extra constants of motion needing to be dealt with in the stat mech.  I think this is probably why the internal energy (i.e. energy in a nonrotating rest frame) is used.  It makes the problem analysis simpler.

David Bowman