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Exact and numerical results for the behavior of finite-dimensional inhomogeneous statistical-mechanical systems exhibiting phase transitions

Project DN 02/8 (ДН 02/8) 2018-2019 with the  National Science Fund of Bulgaria:




tel. (+359 2)


Prof. DSc Daniel Danchev - Project leader


979 6447

Prof. Dr. V. Vassilev (Solid Mechanics)


979 6478

Prof. Dr. P. Djondjorov (Solid Mechanics)


979 6478

Assist. Prof.  Dr. Galin Valchev


979 6701


The current project aims to provide exact results for basic statistical-mechanical models, as well as to obtain numerical results for them. We will study the modifications in the phase diagrams of such systems due to their finite size, the behavior of the order parameters profiles and the response functions. Special attention will be paid to the fluctuation induced interactions, including the Casimir effect, in model fluid systems undergoing phase transitions near the respective critical points of the infinite (bulk) or the finite systems. This topic is very important and currently actively researched since the above-mentioned forces bear a significant impact on the understanding of behavior and manipulation of nano-devices. Currently, the Casimir effect and other similar phenomena are a research object in the quantum electrodynamics, the chromodynamics, the cosmology, the condensed matter physics, in some branches of biology, as well as in nano-technologies.

The interested reader is directed to the following important review papers in this field [1.1-1.5]. With respect to the current knowledge on the critical Casimir effect, whose properties are of main interest in the present project, to a certain degree the main results are summarized and discussed in the following review articles [1.6,1.7], some more specific aspects are discussed in [1.4,1.8,1.9]. As it is becoming clear, the study of the properties of Casimir effect in different fields, unavoidably needs and involves knowledge from mathematics as well as numerical methods, and computer systems.


[1.1] A. Rodriguez, P.-C. Hui, D. Woolf, S. Johnson, M. Lončar and F. Capasso, Classical and fluctuation-induced electromagnetic interactions in micron-scale systems: designer bonding, antibonding, and Casimir forces, Ann. Phys., 527(1-2), 45-80, 2015.

[1.2] G. Klimchitskaya and V. Mostepanenko, Casimir and van der Waals forces: Advances and problems, Proc. of Peter the Great St.Petersburg Polytechnic University, N1(517), 41-65, 2015.

[1.3] L. Woods, D. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A. Rodriguez and R. Podgornik, A materials perspective on Casimir and van der Waals interactions, ArXiv e-prints, 2015.

[1.4] O. Vasilyev, Monte Carlo Simulation of Critical Casimir Forces. Order, Disorder and Criticality, vol. 4, ch. 2, 55-110, World Scientific, 2015.

[1.5] R. Zhao, Y. Luo and J. Pendry, Transformation optics applied to van der Waals interactions, Sci. Bull., 61(1), 59-67, 2016.

[1.6] M. Krech, Casimir Effect in Critical Systems, World Scientific, Singapore, 1994.

[1.7] J. Brankov, D. Dantchev and N. Tonchev, The Theory of Critical Phenomena in Finite-Size Systems – Scaling and Quantum Effects, World Scientific, Singapore, 2000.

[1.8] A. Gambassi and S. Dietrich, Critical Casimir forces steered by patterned substrates, Soft Matter, 7, 1247-1253, 2011.

[1.9] D. Dean, Non-equilibrium fluctuation-induced interactions, Phys. Scripta, 86(5), 058502, 2012.








Modified date:01-03-2019