Post on 09-Mar-2019
transcript
Sommario lezione
Introduzione alla termodinamica relativistica
Temperatura ed entropia in relativita'
Insiemi relativistici
IntroduzioneMotivazioni teoriche e fenomenologiche:
Teoria dei campi quantistici a “temperatura finita” (thermal field theory) Fisica dei plasmi relativistici astrofisici QuarkGluon plasma e le collisioni di ioni pesanti ultrarelativistici Termodinamica nei campi gravitazionali forti (stelle compatte) Processo di adronizzazione nelle collisioni di alta energia
Nei sistemi creati in laboratorio (QGP) si hanno densità finite, ma anche volumi e cariche finite, perciò l'ipotesi di limite termodinamico che si fa usualmente deve essere attentamente valutata
An example: ideal Boltzmann gas
v
?
The moving observer sees momentum distribution function changed. How?
An obvious costraint is the invariance of number of particles
Assume the observer at rest with the gas can write
In relativity, the infinitesimal volume is NOT a scalar, rather a fourvector
where is a threedimensional spacelike hypersurface in Minkowski spacetime
For the observer at rest and therefore
Particle current
invariant
We are then led to demand that be turned into a Lorentz scalar
Under a Lorentz transformation
Note that T and V do not change here because they are meant to be parameters shaping the distribution as seen by the observer at rest.
The observer in motion sees a distribution proportional to
Temperature fourvector
The thermodynamical equilibrium as seen by an inertial observer is described by a temperature fourvector. For an inertial observer at rest with the system (momentum=0)
From now on, when referring to the word “temperature”, the restframe is implied, i.e.The temperature is:
The particle current fourvector reads:
(Grand)Canonical ensemble in quantum statistical mechanics
Maximize entropy with fixed stationary values of some quantities (not necessarilyCommuting with each other):
It can be shown that (R. Balian vol. 1) that – because of the trace one can take the derivativewith respect to the statistical operator as though it was a real variable
Entropy in relativityIn nonrelativistic mechanics, entropy is independent of the overall motion of the gas.It only depends on internal energy, not on total energy (kinetic+internal). In thermodynamicsthis is an assumption, not so in Statistical Mechanics.
PROOF
Maximizing entropy with the constraint of a nonvanishing mean momentum leads to
V being, as yet, a Lagrange multiplier with the dimension of a velocity.
Under a Galileian transformation with v NOTE
Thus, entropy is independent of the collective motion of the system and of its kinetic energyMv2/2.It seems natural to demand the same in relativistic extension, namely entropy should be a Lorentzinvariant quantity
Nonrelativistic physics
Relativistic physics
Canonical ensemble in relativity (traditional approach)
Introduce canonical and grandcanonical ensemble through a reservoir
Reservoir
System
System+reservoir = microcanonical ensembleS+R interaction must be weak and short range(only through contact surface)
Classically
Probability of a state of the system: sum overthe reservoir states
Canonical ensemble in relativity – cont'd
Reservoir
System
Entropy is a function of fourmomentum
Since S = log W is a Lorentz invariant
Relativistic density operatorMaximize entropy with the constraints of energy and momentum conservation leads to
But and so
In relativity, also is an invariant and indeed the traceis a relativistic invariant (sum over states)
Similarly, one can introduce chemical potentials, which are generally related to conservedcharges rather than number of particles. This is simpler to extend as charges are already Lorentz scalars.
Starting from the new form of the Gibbs distribution, the ideal gas distribution functionIn covariant form can be recovered.
Particularly
QUESTION: Why is the partition function invariant under a Lorentz transformation and not in nonrelativistic physics under a Galileian transformation?HINT Take the rest energy term into account and the fact that temperature is afourvector, i.e. its time component is not invariant under Galileian transformation
Relativistic thermodynamics
In the rest frame (of the reservoir subtlety here) they reduce to the usual relation known from classical thermodynamics.
All thermodynamical relations of classical thermodynamics apply to relativistic thermodynamics provided that:
T is interpreted as the proper temperature (measured in the reservoir's rest frame)
V is interpreted as the proper volume (measured in the reservoir's rest frame)
All chemical potentials are relevant to some charge conservation
More on temperature in relativity
Comoving thermometer:Thermodynamic equilibrium of bothenergy and momentum Proper temperature
Thermometer at rest, moving system:Thermodynamic equilibrium ofenergy, but not of momentum. Measured temperature is redshifted
Enforce the equilibrium condition for two bodies in thermal contact (system and thermometer)
●If comoving
●If thermometer at rest
Therefore, the temperature measured by a noncomoving thermometer is notthe proper temperature, but redshifted by a factor 1/g This is the “classical” viewpoint by Einstein and Planck
Classical statistical ensembles Microcanonical: energy and number of particles fixed
Canonical: number of particles is fixed, but energy fluctuates because the system is in contact with a reservoir
Grandcanonical: both number of particles and energy fluctuate
Relativistic statistical ensembles
These definitions are more rigorous. The previous ones are approximate expressionswhich are equivalent for sufficiently large volumes, i.e. when this replacement is possible
One particle in NRQM reexamined
Therefore, using the new definition and inserting a complete set of momentum eigenstates:
So:
Because of the completeness relation in the box:
Therefore:
This is a continuous and derivable function of E !
This is the same expression as in classical mechanics, for a single free particle confined within a box with volume V
For sufficiently large volumes
Microcanonical ensemble: energymomentum and charges (additive) fixed
Canonical: fixed charges, energymomentum fluctuates because in contact with areservoir
Grandcanonical: both charges and energymomentum fluctuatetemperature fourvector
NOTE: partition functions are Lorentz invariants