Vol. 64, No. 8, August 2014, pp. 819∼824 New Physics: Sae Mulli, DOI: 10.3938/NPSM.64.819 Dynamic Scaling Analysis of Critical Behaviors in Nonequilibrium Processes Meesoon Ha * Department of Physics Education, Chosun University, Gwangju 501-759, Korea (Received 16 July 2014 : revised 22 July 2014 : accepted 22 July 2014) We present a method for analyzing the critical behaviors systematically in nonequilibrium pro- cesses by using dynamic scaling, where we extend the well-known finite-size scaling (FSS) theory for the time evolution of major physical quantities that can indicate either a phase transition or some scaling property. Particularly, we discuss two cases: one is the one-dimensional (1D) thin film growth by vapor deposition polymerization (VDP), and the other is the synchronization of globally- coupled oscillators. Using a dynamic scaling analysis, we show that the universality issue of critical behaviors in nonequilibrium processes can be investigated even though the system is neither in the steady-state limit nor in the thermodynamic limit. Finally, in the context of this extended FSS analysis, we compare the VDP growth with the modified 1D Kardar Parisi-Zhang-type growth and classify the characteristics of synchronization transitions with various setups. PACS numbers: 05.10.Ln, 05.40.-a, 02.70.-c, 64.60.-i, 68.55.-a Keywords: Nonequilibrium processes, Dynamic scaling, Finite-size scaling, VDP growth, Synchronization of coupled oscillators I. INTRODUCTION Scaling properties are ubiquitous in nature with real systems far from equilibrium. As tuning the control pa- rameter of the system, it may undergo a nonequilibrium phase transition from one phase to the other one. More- over, it is well known that the system exhibits collective behaviors near and at the criticality, where the corre- lation length becomes diverging and covers all over the system. From theoretical point of view, it is very in- teresting and important how to classify various physical properties of such phase transitions in practice. In order to figure out the thermodynamic limiting behavior of the order parameter in the steady state, one can employ the finite-size-scaling (FSS) theory from equilibrium one. However, in the absence of analytically exact solutions for physical processes, numerical tests are inevitable, which are unavoidably limited to finite systems and com- puting facilities. Such an issue has long been already recognized in the context of phase transitions even in equilibrium that the limitation can be exploited [1] to * E-mail: msha@chosun.ac.kr yield insight into the transition nature and the FSS ef- fect. This concept has been already extended to inves- tigate dynamic phase transitions in nonequilibrium sys- tems [2,3]. When a continuous transition contains some crossover behaviors because of nontrivial finite-size cor- rection to scaling, such a careful FSS analysis is particu- larly valuable since it is governed by the FSS exponent. Although the FSS analysis is a quite powerful tool to resolve continuous phase transitions and the universality, some technical difficulty remains to obtain enough data in reasonable sizes. It is because numerical simulations take quite long CPU time until the system reaches its steady state. The bigger system the more CPU time al- gebraically. Due to this, one might analyze incomplete data in small systems that result in some wrong con- clusion. To avoid such a misleading analysis, a variety of side techniques are available, such as the higher mo- ment analysis of the order parameter. However, most of them still require the steady-state limit. Thus, it is necessary to find a systematic analysis for temporal be- haviors as well as the relaxation of collective behaviors near the saturation at the criticality. This corresponds to the extension of FSS with dynamic scaling [2,3]. 819 This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.