Sistemi relativistici

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1 Sistemi relativistici Corso di laurea magistrale F. Becattini

2 Sommario lezione Introduzione alla termodinamica relativistica Temperatura ed entropia in relativita' Insiemi relativistici

3 Introduzione Motivazioni teoriche e fenomenologiche: Teoria dei campi quantistici a temperatura finita (thermal field theory) Fisica dei plasmi relativistici astrofisici Quark Gluon 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

4 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

5 Assume the observer at rest with the gas can write In relativity, the infinitesimal volume is NOT a scalar, rather a four vector where S is a three dimensional time like hypersurface in Minkowski space time For the observer at rest and therefore invariant Particle current

6 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 four vector

For an inertial observer at rest with the system (momentum=0) From now on,

7 The particle current four vector reads: The thermodynamical equilibrium as seen by an inertial observer is described by a temperature four vector. For an inertial observer at rest with the system (momentum=0) From now on, when referring to the word temperature, the rest frame is implied, i.e. The temperature is:

(Grand )Canonical ensemble in quantum statistical mechanics Maximize entropy with fixed stationary values of some quantities (not necessarily Commuting with each

8 (Grand )Canonical ensemble in quantum statistical mechanics Maximize entropy with fixed stationary values of some quantities (not necessarily Commuting with each other): It can be shown that (R. Balian vol. 1) that because of the trace one can take the derivative with respect to the statistical operator as though it was a real variable

Entropy in relativity In non relativistic mechanics, entropy is

It only depends on internal energy, not on total energy

In thermodynamics this is an assumption, not so in Statistical

9 Entropy in relativity In non relativistic mechanics, entropy is independent of the overall motion of the gas. It only depends on internal energy, not on total energy (kinetic+internal). In thermodynamics this is an assumption, not so in Statistical Mechanics. PROOF Maximizing entropy with the constraint of a non vanishing mean momentum leads to V being, as yet, a Lagrange multiplier. Under a Galileian transformation NOTE

10 Therefore and The equation Independent of v makes it clear the meaning of v ENTROPY

11 Thus, entropy is independent of the collective motion of the system and of its kinetic energy Mv 2 /2. It seems natural to demand the same in relativistic extension, namely entropy should be a Lorentz invariant quantity Non relativistic physics Relativistic physics

Canonical ensemble in relativity (traditional approach) Introduce canonical and grand canonical ensemble through a reservoir System+reservoir = microcanonical ensemble

12 Canonical ensemble in relativity (traditional approach) Introduce canonical and grand canonical ensemble through a reservoir System+reservoir = microcanonical ensemble S+R interaction must be weak and short range (only through contact surface) System Classically Probability of a state of the system: sum over the reservoir states Reservoir

13 Reminder: marginal distribution for the system

14 Canonical ensemble in relativity cont'd Entropy is a function of four momentum System Since S = log W is a Lorentz invariant Reservoir

15 Relativistic density operator Maximize entropy with the constraints of energy and momentum conservation leads to But and so In relativity, also is an invariant and indeed the trace is a relativistic invariant (sum over states)

16 Particularly Similarly, one can introduce chemical potentials, which are generally related to conserved charges 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 function In covariant form can be recovered. QUESTION: Why is the partition function invariant under a Lorentz transformation and not in non relativistic physics under a Galileian transformation? HINT Take the rest energy term into account and the fact that temperature is a four vector, i.e. its time component is not invariant under Galileian transformation

17 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

18 More on temperature in relativity Comoving thermometer: Thermodynamic equilibrium of both energy and momentum Proper temperature Thermometer at rest, moving system: Thermodynamic equilibrium of energy, but not of momentum. Measured temperature is red shifted

temperature measured by a non comoving thermometer is not the proper

19 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 non comoving thermometer is not the proper temperature, but red shifted by a factor 1/γ This is the classical viewpoint by Einstein and Planck

20 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 Grand canonical: both number of particles and energy fluctuate

21 Relativistic statistical ensembles These definitions are more rigorous. The previous ones are approximate expressions which are equivalent for sufficiently large volumes, i.e. when this replacement is possible

22 One particle in NRQM reexamined Therefore, using the new definition and inserting a complete set of momentum eigenstates: So:

23 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

fixed Canonical: fixed charges, energy momentum

canonical: both charges and energy momentum fluctuate

24 For sufficiently large volumes Microcanonical ensemble: energy momentum and charges (additive) fixed Canonical: fixed charges, energy momentum fluctuates because in contact with a reservoir Grand canonical: both charges and energy momentum fluctuate temperature four vector NOTE: partition functions are Lorentz invariants

Sistemi relativistici

Sistemi relativistici Corso di laurea magistrale 2014 15 F. Becattini Sommario lezione Problemi dei sistemi piccoli Trattazione generale dell'insieme microcanonico per un campo libero Problems The large