Transport Theories of Heavy-Ion Collisions

1. Transport Theories of Heavy-Ion Collisions Elena Bratkovskaya 27-31 Aug. 2008 , 20th Indian-Summer School of Physics& 4th HADES Summer School ‚Hadrons in the Nuclear Medium‘

2. Lectures I-III I: From classical dynamics to quantum-field dynamics II: Off-shell relativistic transport theory III: Transport models versus observables Scetch of an ultrarelativistic nucleus-nucleus collision

3. Lecture I: From classical dynamics to quantum-field dynamics

4. Classical dynamics: Newtonian mechanics The dynamical concept is a mathematical formalization for any fixed ‚rule‘ which describes the time dependence of a point's position in its ambient space. The concept of a dynamical system has its origins inNewtonian mechanics. ‚Newton's Second Law‘ (1687): the net force on an object is equal to the mass of the object multiplied by its acceleration , where => Equation of motion (EoM): By solving the EoM, one obtaines the trajectories: r = r(t)

5. Energy in classical dynamics The work done by the force is defined as the scalar product of the force and the displacement vector: • Ek is the kinetic energy:the amount of work done to accelerate the particle from zero velocity to the given velocity u • Let‘s consider only conservative forces F: • defined as the gradient of a potential energyEp • Ep is the work of involved forces to rearrange mutual positions of bodies • Conservation of energy: the total energy is constant in time

6. Momentum in classical dynamics In classical mechanics amomentum is the product of the mass and velocity: Newton's Second Law Linear momentum of a system of particles: Law of conservation of linear momentum: the total momentum of a closed system of objects (which has nointeractions with external fields) is constant. In an isolated system with only two objects, the change in momentum of one object must be equal and opposite to the change in momentum of the other object:

7. Collisions in classical dynamics • Elastic collision of two particles: • conserve kinetic energy as well as total momentum before and after collision • Inelastic collision of two particles: • don't conserve kinetic energy, but the total momentum before and after the collision is conserved • e.g. E.g.: a completely inelastic collision between equal masses(m+m) (2m):

8. Classical dynamics: Lagrangian mechanics Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. (Joseph Louis Lagrange, 1788) • From Newton's law for a system of particles => the total force on each particle is • D'Alembert's principle: the virtual work, δW, done by the total and inertial forces together through an arbitrary virtual displacement, dri, = 0 • Introduce generalized coordinates, qj • generalized forces:

9. Classical dynamics: Lagrangian mechanics The kinetic energy, T, for a system of particles (1) If the Fi are conservative, they may be represented by a gradient of a scalar potential field, V: (2) Eqs. (1) and (2) can be reformulated in terms of the Lagrangian: (Joseph Louis Lagrange, 1788) Lagrangeequations: Hamilton's principle: the system follows the trajectory between t0 and t1 whose action Shas a stationary value , i.e. t1 t0

10. Lagrangian mechanics : generalized coordinates and matching generalized velocities Classical dynamics: Hamiltonian mechanics Hamiltonian mechanics is a re-formulation of classical mechanics that was introduced in 1833 by the Irish mathematician William Rowan Hamilton Hamiltonian mechanics: generalized coordinates and conjugate momentum Each side in the definition ofproduces a differential: matching coefficients Canonical equations of Hamilton: * Hamilton's equations are first-order differential equations, and thus easier to solve than Lagrange's equations, which are second-order.

11. Limits of validity for classical mechanics Newtonian, or non-relativistic classical momentum p=m0u is the result of a first order Taylor approximation of the relativistic expression => classical non-relativistic mechanics is valid only if

12. Classical vs. Quantum mechanics Classical mechanics: Quantum mechanics macro systems micro sytems particle is described by wave packet (X,t) Y(X,t) • wave–particle duality: all matter and energy exhibits both wave-like and particle-like properties Werner Heisenberg & Erwin Schrödinger • Heisenberg uncertainty principle: locating a particle in a small region of space makes the momentum of the particle uncertain; and conversely, that measuring the momentum of a particle precisely makes the position uncertain Planck constant: • in classical physics - all observables commute and the commutator is zero, e.g. [x,p]=0 • in quantum physics - canonical commutation relation, e.g.

13. Quantum mechanics: Schrödinger equation TheSchrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics. ( Erwin Schrödinger, 1926 ) For a general quantum system: the wavefunctionis the probability amplitude for different configurations of the system the Hamiltonian operator For a single particle in three dimensions: is the wavefunction, which is the amplitude for the particle to have a given position r=(x,y,z) at any given time t; m is the mass of the particle; V( r) is the potential energy of the particle at position r • The time-dependent Schrödinger equation describes the time evolution of the quantum state of a single non-relativistic particle

14. d λ2 λ1 Quantum mechanics: EoM Fourier transform of the time dependent Schrodinger equation  Schrödinger equation in momentum representation: Hamilton‘s equations of motion for a single non-relativistic particle : Correspondence principle:(Niels Bohr, 1923) If the wavelength (λ=h/p) of the wavepacket is small compared to the considered scale d, the Schrödinger EoMreproduces the classical EoM. E.g.: l2 ~ d l1 <<d

15. Quantum mechanical description of the many-body system Dynamics of heavy-ion collisions is a many-body problem! Schrödinger equationfor the system of N particlesin three dimensions: • Hartree-Fock approximation: • many-body wave function  • antisym. product of single-particle wave functions • many-body Hamiltonian  single-particle Hartree-Fock Hamiltonian kinetic term N-body potential kinetic term 2-body potential

16. Hartree term: • self-generated local mean-field potential (classical) • Fock term: • non-local mean-field exchange potential (quantum statistics) Hartree-Fock equation Time-dependent Hartree-Fock equation for a single particle i: Single-particle Hartree-Fock Hamiltonian operator: • TDHF approximation describes only the interactions of particles with the time-dependent mean-field UHF(r,t)! • EoM: propagation of particles in the self-generated mean-field • In order to describe the collisions between the individual(!) particles, one has to gobeyond the mean-field level !

17. Density-matrix formalism In order to go beyond the one-body TDHF limit one has to include N-body operators (or at least 2-body operators) density-matrix formalism Schrödinger equation for a system of N fermions: Hamiltonian operator: • Introduce the N-particle density operator: Schrödinger eq. in density matrix representation  von Neumann eq.: Reduced density matrices(tensor of rank 2n), n<N : Integrate out particles n+1…N Normalization:

18. Density matrix formalism: BBGKY-Hierarchy BBGKY-Hierarchy (Bogolyubov, Born, Green, Kirkwood and Yvon) For n=1: and n=2: n<N Correlation dynamics: 1-body density matrix 2-body density matrix: 2PI= 2-particle-irreducible approach 1PI = 1-particle-irreducible approach = TDHF approximation 2-body correlations

19. Correlation dynamics 2-body correlations TDHF EoM: V=(12) – in-medium interaction , i.e. including Pauli-blocking operator Time evolution of c2 depends on the distribution of a third particle, which is integrated out in the trace! Note that the third particle is interacting as well!

20. Vlasov equation (1) BBGKY-Hierarchie - 1PI: 0 • perform Wigner transformation:(Fourier transformation in x-x‘) one-body phase-space distribution function density in coord. space: density in momentum space:

21. Vlasov equation (2) Using Perform a Taylor expansion of gradU around s=0; ! Assumption:keep the 1st term ~ classical limit ! Vlasov equation:  Vlasov eq. is entirely classical !

22. Vlasov equation (3)  Classical equation of motion: Vlasov equation describes the propagation of particles in the time-dependent Hartree-Fock mean-field ! Where is the ‚quantum mechanics‘ ?! The quantum physics plays only a role in the initial conditions for the phase-space distribution function r(r,p,t). The initial r(r,p,t) must respect the Pauli principle, but the identity of the particles plays no role beyond the initial conditions!

23. Uehling-Uhlenbeck equation: collision term I TDHF –Vlasov eq. Collision term: Formally solve the EoM (with some approximations in momentum space): and insert in the expression for I(11‘,t): 

24. Ueling-Uhlenbeck equation: collision term II 1+23+4 Loss term Gain term using for the differential cross section: Collision term =‚Gain‘ term - ‚Loss‘ term particle 1(+2) 3+41+2 1+23+4 This collision term describes collisions of particle 1 (momentum p1) with all possible particles 2 (momentum p2) – obeying energy-momentum conservation. Loss term: 1+23+4, provided that the final states p3 and p4 are not occupied ! The time-reversed processes 3+41+2 are desribed by the gain-term!

25. Vlasov-Uehling-Uhlenbeck (VUU) equation Transform back to phase-space (r,p) and add to Vlasov equation:  VUU or BUU gain loss Here the delta-functions have been integrated out and give the relative velocity v12 of the colliding particles: The VUU equations describes the propagation in the self-generated mean-field U(r,t) as well as mutual two-body interactions respecting the Pauli-principle (also denoted as BUU ect.).

26. Numerical solution of the BUU equation 1) The Vlasov part is solved in the testparticle approximation with A denoting the number of testparticles per nucleon (Ainfinity). The trajectories ri(t), pi(t) result from the solution of classical equations of motion. • 2) The collision term is solved by a Monte Carlo treatment of collisions: • an interaction takes place at impact parameter b if pb2 < s! • the final state is selected by Monte Carlo according to the angular distribution ds/dW • the final state is accepted again by Monte Carlo according to the probability for 1+23+4

27. From classical dynamics to quantum-field dynamics