Koppe's Work of 1948: A fundamental for non-equilibrium rate of particle production
aa r X i v : . [ nu c l - t h ] D ec ECTP-2013-13 and WLCAPP-2013-10
Koppe’s Work of 1948 ∗ :A fundamental for non-equilibrium rate of particle production Abdel Nasser TAWFIK † Egyptian Center for Theoretical Physics (ECTP),MTI University, 11571 Cairo, Egypt andWorld Laboratory for Cosmology And Particle Physics (WLCAPP), Cairo, Egypt (Dated: July 23, 2018)
Abstract
In 1948, Koppe formulated an almost complete recipe for statistical-thermal models including parti-cle production, formation and decay of resonances, temporal and thermal evolution of the interactingsystem, statistical approaches and equilibrium condition in final state of the nuclear interaction. Asthe rate of particle production was one of the basic assumptions, recalling Koppe’s work would be anessential input to be involved in the statistical prediction of non-equilibrium particle production inrecent and future ultra-relativistic collisions.
PACS numbers: 64.60.F-,05.10.GgKeywords: Equilibrium properties near critical points, Fokker-Planck equation in statistical physics, ∗ H. Koppe,
Die Mesonenausbeute beim Beschuss von leichten Kernen mit Alpha-Teilchen , Z. Naturforsch. ,231-232 (1948);[English Translation] Meson Yields from Bombarding Light Nuclei with Alpha Particles, Z. Naturforsch. A xx , xxx (2013) † . RATE OF PARTICLE PRODUCTION IN THERMAL MEDIUM At Betatron energy, the temperature of excited nucleus T = 3 . √ N , where N refersto the number of excited nuclei (resonances), was related to the excitation energy U = m / ( m + m ) E , with m ( m ) and E being mass of projectile (mass of target) nucleus andkinetic energy, respectively. T was measured as ∼
10 MeV. Koppe assumed that even theprojectile ( α -particle) can not remain stable [1]. Therefore, pair production is likely [2]. ”pair-degeneracy” [2] or ”Vacuum dissociation” [3] was principally investigated from electron-positronpair production. Koppe assumed that the same considerations make it possible to apply thisin meson-pair production [1] and expected a very small number of produced mesons.The electron (produced particle) gas can be treated as cavity radiation with a concrete energydensity relating radiation loss to cross-section of excited nuclei σ . The temporal evolution oftemperature should reflect the expansion of the interacting system. The speed of electrons(produced particles) is very close to c . Then, the energy flux caused by electrons relative tolight quantum radiation will be increased by the same factor 7 /
8. The rate of produced particlewas calculated as, ν ( T ( t )) = m b σπ ¯ h T ( t ) e − mb c T ( t ) . (1)Then, the integration results in n = a ( m + m ) T exp( − m b c /T ), where a = 0 . T , then the number of mesons which should be producedin α − A collisions at 380 MeV per unit time was estimated as ∼ . × − . II. PARTICLE PRODUCTION AND NON-EQUILIBRIUM PARTICLE DISTRIBU-TION
In ultra-relativistic nuclear collisions, deconfinement and/or chiral broken-symmetry restora-tion phase transition(s) is(are) supposed to take place. To study the dynamics and velocitydistribution of objects in such thermal background, like transport properties in quark-gluonplasma, Fokker-Planck equation is a well-known tool. The statistical properties of an ensembleconsisting of individual parton objects is given by non-equilibrium single-particle distributionfunction f [4–6]. The probability of finding an object in infinitesimal region in phase spaceis directly proportional to the volume element and f . The latter is assumed to fulfil the2oltzmann-Vlasov (BV) master equation,˙ f + ˙ ~x · ∇ x f + ˙ ~k · ∇ k f + ˙ ~q c · ∇ q c f = G + L . (2)The first term in rhs G represents gain or rate of particle production with momentum k + kt ,which is conjectured to lose momentum kt due to reactions with the background. The secondterm L represents loss due to the scattering rate. The effective potential U has to combinethe well-known Coulomb U ( x → ∝ /x and confined potentials U ( x → ∞ ) ∝
0. Thestandard position ~x and momentum ~k variables are given in first two terms in lhs. The thirdterm represents the dynamics of the charge, where ˙ ~k can be given by field tensor. The fourthterm reflects an extension of phase space to include color charge, ˙ ~q c .Studying the stochastic behavior of a single object propagating with random noise knownas Langevin equation represents one way to solve this problem. A master equation, such asthe linearised BV equation, with the Landau soft-scattering approximation would give anothermethod. Koppe’s work would be a fundamental-statistical approach for a qualitative estimationfor the non-equilibrium rate of particle production. [1] H. Koppe, Z. Naturforschg. , 251-252 (1948).[2] H. Koppe, Ann. Phys.(Berlin) , 103-112 (1948).[3] F. Houtermans and H. Jensen, Z. Naturforsch. , 146 (1947).[4] A. Tawfik, Int. J. Theor. Phys. , 1396-1407 (2012).[5] A. Tawfik, J.Phys. G40 , 055109 (2013).[6] A. Tawfik, Nucl. Phys. A , 63-72 (2011)., 63-72 (2011).