Francesco Becattini, University of Florence

Hydrodynamics of fluids with spinHydrodynamics of fluids with spin

F. B., F. Piccinini, Ann. Phys. 323, 2452 (2008). F.B., L. Tinti, arXiv:0911.0864, to appear (hopefully soon) in Ann. Phys.

OUTLINEOUTLINE

Introduction Relativistic fluids at thermodynamical equilibrium The meaning of the spin tensor Ongoing investigations

Heavy ion forum, CERN, July 9 2010 MOTIVATIONS

The success of the hydrodynamical description of Quark Gluon Plasma

Reaction z plane

y

x

What is a fluid with spin ? What is a fluid with spin ?

A fluid with spin is a fluid having a macroscopic spin density and which thus needs a spin tensor to be described, besides the stress­energy tensor

Why should QGP be a fluid with spin?Why should QGP be a fluid with spin?

If relaxation times are short, also spin degrees of freedom could locally equilibrate quickly and we could end up with a fluid a finite (though small) spin density, particularly in peripheral collisions A short history of relativistic fluids with spinA short history of relativistic fluids with spin

M. Mathisson, Acta Phys. Pol. 6, 163 (1937).

J. Weyssenhoff, A. Raabe, Acta Phys. Pol. 9, 7 (1947).

Frenkel condition

D. Bohm and J.P. Vigier, Phys. Rev. 109, 1882 (1958) Critique (well founded) of the Frenkel condition

F. Halbwachs, Theorie relativiste des fluids `a spin, Gauthier­Villars, Paris, (1960) Variational lagrangian theory of ideal fluids with spin, still embodying Frenkel condition

From 1960 onwards: many papers with extension and possible applications of Halbwach's theory to general relativity, Einstein­ Cartan theory and relevant cosmologies

The simplest fluid: The simplest fluid: rigidly rotating relativistic ideal Boltzmann gasrigidly rotating relativistic ideal Boltzmann gas

This is a system at complete thermodynamical equilibrium and can be used to learn much about the hydrodynamics of fluids with spin without begging the question, i.e. without introducing any prior stress­energy and spin tensor. The only assumption is statistical equilibrium.

For large enough systems:

Grand­canonical rotational partition function

which can be obtained by maximizing with the constraints of energy and angular­momentum conservation Why is it rigidly rotating? Landau's argument Why is it rigidly rotating? Landau's argument

E , i Pi

Maximize entropy with constraints

Local temperature

Rotating fluid at thermodynamical equilibrium: entropyRotating fluid at thermodynamical equilibrium: entropy

As a consequence of

with

we obtain

For an ideal Boltzmann gas

Rotating gas at thermodynamical equilibrium: polarizationRotating gas at thermodynamical equilibrium: polarization F. B., F. Piccinini, Ann. Phys. 323, 2452 (2008).

in the inertial frame

This polarization is very small for usual systems but it is just what causes Barnett effect (see later)

Crucial steps in the derivation

To obtain the polarization expression, the key ingredient is the matrix element

which in turn stems from an ansatz

motivated by the enforcement of

However, the same expression can be obtained by maximing the entropy as a functional of phase space distribution (in progress)

From global to local: hydrodynamical viewFrom global to local: hydrodynamical view

The spinning ideal gas, if large, can be seen as a fluid and local quantities, like proper energy density or pressure can be inferred from the expression of Z 

Since particles are polarized, there must exist also a spin tensor:

Local thermodynamical relation

To make it covariant the angular velocity is to be replaced with the acceleration tensor constructed with the Frenet­Serret tetrad

See F. Halbwachs “Theorie relativiste des fluids a spin”, Paris, 1960

The acceleration tensor

In the inertial frame

For a rigidly rotating body

Already been used in literature in GR and Einstein­Cartan theory. It should be stressed that the last term is a QUANTUM correction (vanishes if h=0) to the otherwise familiar thermodynamical relation For the ideal Boltzman gas, one can derive the expression of the phase­space distribution (F.B., L. Tinti, arXiv:0911.0864, Ann. Phys. in press)

or, in Dirac spinor's notation

Spin density tensor for an ideal Boltzmann gasSpin density tensor for an ideal Boltzmann gas

This implies that the Frenkel condition is VIOLATED

even for the simplest fluid with spin!

This deserves a little reflection. It means that we cannot describe the spin density of a fluid with ONE spacial vector (polarization), but we need TWO.

This is unfamiliar because a single quantum relativistic particle has t=0 but a fluid is made of many particles with random polarization vectors

Does the spin tensor have a direct physical meaning?

Definition of macroscopic tensors with the correspondance principle

In classical and quantum field theory, these tensors are somewhat arbitrary and can be changed, provided that they fulfill the same conservation equations and that they yield the same total energy, momentum and angular momentum

However, there is also thermodynamics, which dictates a definite relation between mechanical (pressure, energy, related to stress­energy tensor) and statistical quantities (entropy, i.e. the number of degrees of freedom, related to the discrete nature of matter). This can help to single out one couple, or at least to constrain it.

Is the spin tensor arbitrary?

From pure thermodynamics, we have inferred that there is a non­vanishing polarization of particles for the simplest fluid at equilibrium with finite angular momentum and this in turn implies that spin density is non­vanishing as well

Therefore, if we want to keep the correspondance between microscopic quantum and macroscopic tensors

the only possible conclusion is that NOT ALL transformation of stress­energy and spin tensors are allowed.

Particularly, the Belinfante symmetrization procedure seems to be ruled out by thermodynamics

From a local point of view, thermodynamics enforces a relation between the stress­energy tensor and a quantity which is beyond a classical relativistic theory: ENTROPY.

Entropy can only be introduced counting states, i.e. in a quantum field framework; it requires the existence of particles.

EXPERIMENTAL EVIDENCE: Barnett effectEXPERIMENTAL EVIDENCE: Barnett effect S. J. Barnett, Magnetization by Rotation, Phys. Rev. 6, 239–270 (1915).

See also E. Purcell, Astroph. J. 231, 404 (1979)

Spontaneous magnetization of an uncharged body when spun around its axis

It is a dissipative transformation of the orbital angular momentum into spin of the constituents. The angular velocity decreases and a small magnetic field appears; this phenomenon is accompanied by a heating of the sample.

EXPERIMENTAL EVIDENCE: Barnett effectEXPERIMENTAL EVIDENCE: Barnett effect S. J. Barnett, Magnetization by Rotation, Phys. Rev. 6, 239–270 (1915).

See also E. Purcell, Astroph. J. 231, 404 (1979)

Spontaneous magnetization of an uncharged body when spun around its axis

It is a dissipative transformation of the orbital angular momentum into spin of the constituents. The angular velocity decreases and a small magnetic field appears; this phenomenon is accompanied by a heating of the sample.

EXPERIMENTAL EVIDENCE: Einstein­De Haas effectEXPERIMENTAL EVIDENCE: Einstein­De Haas effect

A. Einstein, W. J. de Haas, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings, 18 I, 696­711 (1915)

Rotation of a ferromagnet originally at rest when put into an external H field

An effect of angular momentum conservation and 2nd law: spins gets aligned with H (irreversibly) and this must be compensated by a on overall orbital angular momentum

How to explain microscopically Barnett effect?

Requires transformation of orbital angular momentum into spin

From a collisional­kinetic point of view, this is possible only if particles interact with a spin­orbit coupling

Such an interaction requires an extension of the Boltzmann kinetic equation (see Landau­Lifshitz, Kinetic physics) because collisions, in the usual approximation, are assumed to occur at a fixed space­time point, therefore orbital angular momentum is conserved by construction.

We thus need a non­local collisional integral to include spin­orbit interaction, and this is likely to lead to interesting consequences on the form of the stress­energy tensor (no longer symmetric except at full thermodynamical equilibrium)

Ongoing studiesOngoing studies

Determination of a class of transformations of the stress­energy and spin tensor leaving entropy density (i.e. thermodynamics) invariant

Determination of kinetic coefficients for spin generation and transport and relevant modifications of the stress­energy tensor

Is it possible to transform macroscopic spin into orbital angular momentum without dissipation (the problem of ideal fluid with spin)?

General formulation of hydrodynamics with spin

Possible application to cold atoms ?Possible application to cold atoms ?

J. Thomas, arXiv: 0907.0140

SUMMARY AND OUTLOOKSUMMARY AND OUTLOOK We have calculated the expression of the spin density tensor in a rotating relativistic Boltzmann gas in full thermodynamical equilibrium. We recovered an “assumed” result, i.e. the local entropy density gets an additional term of quantum nature depending on the acceleration tensor.

We have argued that thermodynamics requires the inequivalence of an otherwise equivalent class of stress­energy and spin tensors.

We aim at obtaining a general formulation of relativistic hydrodynamics with spin and study its consequences (applying to QGP). This will involve new kinetic coefficients (spin transport and generation) and a generalization of stress­energy tensor with further dissipative terms. J. Thomas, arXiv: 0907.0140