Statistical physics of nonequilibrium systems : Percolation, voter model, and fluctuation theorem = 비평형계의 통계물리: 침투, 투표자 모형, 요동 정리

다국어 초록 (Multilingual Abstract)

In standard statistical physics focus has been put on equilibrium systems. The assumption of equilibrium for a given material provides us a great advantage in understanding underlying physics. In the formalism of canonical ensemble which is the most convenient to describe statistical physics systems, the system is being in contact with thermal resorvior and microstates satisfy detailed balance condition with the Boltzman probability distribution. Through the use of the partition function one can compute equilibrium statistical physics quantities: Helmholtz free energy, the internal energy, the specific heat, and so forth. Even the phase transition manifested in the form of the singularity of free energy, can also be understood analytically and numerically. By obtaining the critical exponents, the critical phenomena of systems can be grouped into universality classes due to the scaling relations.
In the real world, however, most of systems are not in equilibrium. Due to the timevarying external environment or the existence of flux to and from other systems, the microstates of the system may not be balanced so that the probability of each state is not stationary in time. In nonequilibrium, many approaches which are well-defined in equilibrium fail to describe the system although the nonequilibirum phase transitions can be defined and observed in many systems. Similarly to equilibrium systems, it is believed that nonequilibrium critical behaviors can also be understood by the concept of universality class and that the equilibrium approaches near the criticality can be applied to estimate the critical exponents. Through Chapters 2 and 3, we investigate the nature of phase transition of nonequilibrium systems.
In recent years, the reinterpretation of the thermodynamic second law has been actively carried out. For the system in the time-varying external protocol, the work during protocol and the free energy difference between the initial and final equilibrium states can be related by the equality, whereas the thermodynamic second law takes the form of inequality. It is called the fluctuation theorem. In Chapter 4, the application of the fluctuation theorem to a simple physical system is investigated.
In Chapter 2 Percolation properties of growing networks under an Achlioptas process, we study the percolation transition in growing networks under an Achlioptas process (AP). At each time step, a node is added in the network and, with probability δ, a link is formed between two nodes chosen by an AP. We find that there occurs the percolation transition with varying δ and the critical point δ c = 0.5149(1) is determined from the power-law behavior of the order parameter and the crossing of the fourth-order cumulant at the critical point, also confirmed by the movement of the peak positions of the second largest cluster size. Using the finite-size scaling analysis, we get β/ν = 0.20(1) and 1/ν = 0.40(1), which implies β ≈ 1/2 and ν ≈ 5/2. The Fisher exponent τ = 2.24(1) for the cluster size distribution is obtained and shown to satisfy the hyperscaling relation.
We investigate the nature of the phase transition of the coevolving voter model composed of conformists and contrarians in Chapter 3 Phase transition in a coevolving network of conformist and contrarian voters. In the coevolving voter model, each voter has one of two diametrically opposite opinions, and a voter encountering a neighbor with the opposite opinion may either adopt it or rewire the connection to another randomly chosen voter sharing the same opinion. As we smoothly change the relative frequency of rewiring compared to that of adoption, there occurs a phase transition between an active phase and a frozen phase. By performing extensive Monte Carlo calculations, we show that the phase transition is characterized by critical exponents β = 0.54(1) and ν = 1.5(1), which differ from the existing mean-field-typed prediction. We furthermore extend the model by introducing a contrarian type that tries to have neighbors with the opposite opinion, and show that the critical behavior still belongs to the same universality class irrespective of such contrarians’ fraction.
In Chapter 4 Nonequilbrium work by charge control in a Josephson junction, we consider a single Josephson junction in the presence of time varying gate charge, and examine the nonequilibrium work done by the charge control in the framework of fluctuation theorems. Assuming first a high quality junction with negligible Ohmic current, we obtain the probability distribution functions of the work and confirm the Crooks relation to give the estimation of the free energy changes ∆F = 0. The reliability of ∆F estimated from the Jarzynksi equality is crucially dependent on protocol parameters, while the Bennett’s acceptance ratio method yields consistently ∆F = 0. We examine the behaviors of the work average and point out its relation to heat and entropy production associated with the circuit control. Considering finite tunnel resistance we discuss dissipation effects on the work statistics.

목차 (Table of Contents)

• Abstract
• 1 Introduction
• 1.1 Percolation in Erdos and Renyi network
• 1.2 Finite-size scaling
• 1.3 Jarzynski relation
• 2 Percolation properties of growing networks under an Achlioptas process
• 2.1 Introduction
• 2.2 Growing via an Achlioptas process
• 2.3 Results
• 2.4 Summary and Discussion
• 3 Phase transition in a coevolving network of conformist and contrarian voters
• 3.1 Introduction
• 3.2 Coevolving network of voters
• 3.3 Results
• 3.4 Summary
• 4 Nonequilbrium work by charge control in a Josephson junction
• 4.1 Introduction
• 4.2 System and work
• 4.3 PDFs of the work and the Crooks relation
• 4.4 Free energy estimation from the Jarzynski equality and the Bennett method
• 4.5 Work average and entropy production
• 4.6 Dissipation effects
• 4.7 Summary
• 5 Conclusion
• References
• 초록