Critical Behaviour of Directed Percolation Process in the Presence of Compressible Velocity Field

Critical Behaviour of Directed Percolation Process in the Presence of Compressible Velocity Field Master’s Thesis (60 HE credits) Author: Viktor Skultétyˇ † Supervisors: Juha Honkonen, Thors Hans Hansson Affiliation: Fysikum, Stockholm University, Nordic Institute for Theoretical Physics (NORDITA) 2017-04-21, Stockholm †[email protected] secret message Abstract Renormalization group analysis is a useful tool for studying critical behaviour of stochastic systems. In this thesis, field-theoretic renormalization group will be applied to the scalar model representing directed percolation, known as Gribov model, in presence of the random velocity field. Turbulent mixing will be modelled by the compressible form of stochastic Navier-Stokes equation where the compressibility is described by an additional field related to the density. The task will be to find corresponding scaling properties. Acknowledgements First, I would like to thank my supervisors Juha Honkonen and Thors Hans Hansson for great supervision. I express my gratitude to people from the Department of Physics at the University of Helsinki where I have spent a couple of months. I would also like to thank Paolo Muratore-Ginanneschi from the Department of Mathematics for his time and willingness to always help me with my questions. Furthermore, I wish to thank people from the Nordic Institute for Theoretical Physics, especially to Erik Aurell, Ralf Eichhorn for their kind hospitality. My special thanks goes to TomáˇsLuˇcivjanskýfrom Department of Theoretical Physics at Pavol Jozef SafárikUniversityˇ in Koˇsice,who was always willing to give me advices on how to approach the problems I stumbled upon during my work. Finally, I would like to thank my family and friends which were supporting me all the time and without whom this thesis would not be possible. Stockholm, April 2017 Viktor Skultétyˇ Contents Introduction ix 1 Field-Theory approaches to stochastic processes1 1.1 Scaling in the theory of static critical phenomena..............2 1.2 Perturbation theory, Renormalization group.................4 1.2.1 Connected correlation function....................6 1.2.2 Saddle-point approximation, vertex functions.............8 1.2.3 Renormalization.............................9 1.2.4 Anomalies in scale invariance..................... 13 1.3 Critical dynamics................................ 14 1.3.1 Iterative solution to the Langevin equation.............. 16 1.3.2 De Dominicis-Janssen action functional................ 17 1.3.3 Dynamical Renormalization group................... 19 1.4 Reaction-diffusion processes.......................... 21 1.4.1 Doi's second-quantized representation................. 22 1.4.2 Single species annihilation reaction-diffusion process......... 23 1.4.3 Observables and coherent state path integral formalism....... 25 2 Directed percolation 27 2.1 What is directed percolation?......................... 28 2.2 Critical exponents................................ 29 2.2.1 Rapidity symmetry........................... 31 2.2.2 Correlation length and correlation functions............. 31 2.2.3 Scale invariance............................. 32 2.2.4 Dynamic scaling, pair-connectedness function............ 34 2.3 Directed percolation universality class..................... 37 2.3.1 Field-theoretic formulation of DP................... 37 2.3.2 Mean-field theory............................ 39 2.3.3 The renormalization group approach................. 40 2.3.4 Experimental realization of DP.................... 41 3 Turbulent mixing 43 3.1 Basics of fluid dynamics............................ 44 3.1.1 Dissipation............................... 44 3.1.2 Reynolds and Mach number...................... 45 3.2 Incompressible 3D turbulence......................... 46 3.2.1 Fully developed turbulence....................... 46 3.2.2 Kinetic energy dissipation....................... 48 3.2.3 KO41 theory.............................. 50 vii viii Contents 3.3 Compressible 3D turbulence.......................... 52 3.3.1 Kinetic energy dissipation....................... 52 3.3.2 Energy spectra............................. 53 3.4 Field-theoretic formulation of turbulence................... 55 3.4.1 Incompressible Navier-Stokes equation................ 56 3.4.2 Compressible Navier-Stokes equation................. 58 3.4.3 Turbulent diffusion........................... 59 4 Directed percolation process advected by compressible turbulent fluid 61 4.1 Definition of the model, perturbation theory................. 63 4.2 UV divergences, the renormalization group.................. 65 4.2.1 Renormalization of the velocity field................. 66 4.2.2 Renormalization of percolation field.................. 68 4.3 Asymptotic behaviour............................. 70 4.3.1 Stable fixed points of compressible NS model............. 71 4.3.2 Stable fixed points of advected DP process.............. 74 4.3.3 Critical scaling............................. 80 5 Conclusion 83 Appendices 85 A Derivation of propagators and vertices 87 A.1 Propagators................................... 87 A.2 Vertices..................................... 89 B Explicit form of Feynman diagrams 91 B.1 Feynman diagrams for the compressible NS model.............. 91 B.2 Diagrams for the DP model.......................... 92 C Explicit results 95 C.1 Some formulas.................................. 95 C.2 Renormalization constants........................... 95 C.3 Anomalous dimensions............................. 96 C.4 Beta functions.................................. 97 C.4.1 Beta functions for compressible NS.................. 97 C.4.2 Beta functions for DP......................... 99 Bibliography 101 viii Introduction The foundation of quantum field theory In 1928 Paul Dirac postulated a relativistic equation of motion for the electron [1]. This led to an enormous development in theoretical physics and to the formulation of quantum field theory. Historically, the first theory to be constructed was quantum electrodynamics which describes the electromagnetic interactions between electrons, positrons and pho- tons. A new theoretical framework became available for the perturbative calculation of particle scattering processes in high energy physics. Despite its early successes, quantum field theory was plagued by several serious technical difficulties. While the calculation of the low orders of perturbation theory was quite successful, higher order approximations were giving divergent contributions. These divergences, which arise from the large and small momentum scales, were clearly unphysical, since the measurements were showing finite results. It later also appeared that this is not only the problem of quantum electrodynamics, but the problem of quantum field theories in general. The problem of infinities was approached by many physicists and solved in the late 1940s mainly by Feynman, Schwinger, Tomonaga, Dyson [2]. They invented a method of eliminating these divergences without destroying the physical meaning of the theory. This method is known as renormalization. After the divergences were successfully eliminated, the discovery of the running cou- pling constant was made. For example the electric charge e was no longer a constant but its exact value depended on the scale at which it was measured. This led to the important concept of the renormalization group, which was constructed in a solid form in the mid 1950s by Bogoliubov and Shirkov based on the work of previous authors [2,3]. From a practical point of view, the renormalization group technique is an effective method of calculating correlation functions at large or small momentum scales (or equivalently at small or large spatial scales). About the same time, a different field of physics was dealing with another unsolved problem related to scale invariance. Roughly speaking, scale invariance is a feature of objects or laws that do not change if length scales or other scales are multiplied by a com- mon factor. These structures typically obey a non-trivial power-law scaling behaviour. In statistical physics, scale invariance is an important property of a system that un- dergoes certain phase transitions. Namely, so called second order phase transitions are characterized by a divergent correlation length which forces the system to be in a unique scale-invariant state. Measurements show [2] that various different systems have the same large-scale properties, while undergoing this type of phase transition. Properties such as power laws appear to be universal and independent of the microscopic structure of the system. This led to the concept of universality classes, which groups various systems with identical large-scale properties together, depending only on their general character- istics such as symmetry of the system, dimensionality of the space etc. In 1971 Wilson ix x Contents introduced a systematic renormalization procedure based on eliminating the microscopic degrees of freedom [4,5]. Starting from a path integral formulation of the quantum field theory he explained the existence of universality. This was the first appearance of path integral renormalization group in statistical physics. Even though the renormalization group was initially developed for solving problems in particle physics and critical phenomena, it was later realized to be useful also for solving other problems, such as dynamical processes. Field theory approach to dynamical processes The term dynamics in a broad sense refers to any system evolving in time. General properties of macroscopic systems are almost

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