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1

The Monte Carlo Computer Experiment to Study the Order and Phase Transitions in the Mixed Phase Region Based on the Example of the 3D Ashkin-Teller Model

Wojtkowiak Z., Musiał G.

Computational Methods in Science and Technology

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2021

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Vol. 27, No. 3

129--136

EN

In this paper, we demonstrate a new way of performing Monte Carlo (MC) simulations in a mixed phase region that is difficult to study, where with certain probabilities there are different ordering ways. That results in a large oscillation of the values of the computed thermodynamic quantities, which makes their interpretation very problematic. Our results are presented on the example of the 3D Askin-Teller (AT) model, where within a certain range of parameters with equal probabilities there are two different, but equivalent, ways of ordering two of the three order parameters showing independent behavior. The use of our new approach in an MC computer experiment allowed us to use Binder cumulant as well as Challaand the Lee-Kosterlitz-like cumulants. This made it possible to locate phase transitions precisely enough to be able to use the energy distribution histogram method. According to the most effective strategy in the critical region we use our recently proposed cluster MC algorithm and the Metropolis algorithm beyond it, which are suitable for both the first-order and the continuous phase transitions in the 3D AT model. The new approach was demonstrated by determining smooth curves of magnetization and internal energy, and as a consequence by determining the location and character of the phase transition on the line between the mixed phase region and the paramagnetic phase.

2

Computation of latent heat in the system of multi-component order parameter : 3D Ashkin-Teller model

Jeziorek-Knioła D., Wojtkowiak Z., Musiał G.

Computational Methods in Science and Technology

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2019

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Vol. 25, No. 1

31--39

EN

The method for computing the latent heat in a system with many independently behaving components of the order parameter proposed previously is presented for a chosen point of the phase diagram of the 3D Ashkin-Teller (AH) model. Binder, Challa, and Lee-Kosterlitz cumulants are exploited and supplemented by the use of the energy distribution histogram. The proposed computer experiments using the Metropolis algorithm calculate the cumulants in question, the internal energy and its partial contributions as well as the energy distribution for the model Hamiltonian and its components. The important part of our paper is an attempt to validate the results obtained by several independent methods.

3

Distributed Processing of the Lattice in Monte Carlo Simulations of the Ising Type Spin Model

Murawski S., Musiał G., Pawłowski G.

Computational Methods in Science and Technology

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2015

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Vol. 21, No. 3

117--121

EN

Parallelization of processing in Monte Carlo simulations of the Ising spin system with the lattice distributed in a stripe way is proposed. Message passing is applied and one-sided MPI communication with the MPI memory window is exploited. The 2D Ising spin lattice model is taken for testing purposes. The scalability of processing in our simulations is tested in real-life computing on high performance multicomputers and discussed on the basis of speedup and efficiency. The larger the lattice the better scalability is obtained.

4

Parallel Simulations of the Monte Carlo Type: 3D Ashkin-Teller Model

Musiał G., Dębski L., Kamieniarz G.

Computational Methods in Science and Technology

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2007

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Vol. 13, nr 1

41-46

EN

The method of parallelization of processing in simulations of the Monte Carlo type is proposed and tested on the 3D Ashkin- -Teller model characterized by a rich phase diagram. The message passing is applied and the MPI library is exploited. It is demonstrated that the method works well in different regions of the phase diagram, where the phase transitions are first-order and continuous. The dependence of speedup and efficiency on the number of parallel processes is studied. The condition for the best speedup and efficiency in these simulations is formulated and discussed using the results obtained on symmetric NUMA multiprocessor and on symmetric multicomputer. The suggestion as to effective use of the method even in highly heterogeneous computer systems is also given.

5

Alojzy Melich - życie i praca

Musiał G.

Kwartalnik Historii Nauki i Techniki

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2003

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R. 48, nr 1/2

57--82

EN

The article discusses the life and work of Alojzy Melich, professor (doctor habilitatus) of economic sciences the Higher School of Economics (sińce 1974 the Karol Adamiecki Economic Academy) in Katowice. Alojzy Melich was the first to receive a doctoral degree at that tertiary school and has been connected with it until now. In the years 1964-1988 he was Head of the Institute of Political Economy, and in the years 1965-1975 he was President o f the Higher School of Economics (Karol Adamiecki Economic Academy) in Katowice. In 1971 he was nominated a corresponding (associate) memberof the Polish Academy of Sciences, and in the same year he was granted an honorary doctorate by the Martin Luther University in Halle-Wittenberg. Profesor Alojzy Melich’s concern with economics as a branch science focuses on issues of wages and distribution. He has devised an original system of collective wages, which has been put into practice in many branches of industry in Poland. Professor Melich has written over 300 works, including 28 books on economics and about 120 studies, with the rest consisting of papers and articles.

6

Verification of the three-cluster molecular – field renormalization group approach for the ising model

Kamieniarz G., Musiał G.

Computational Methods in Science and Technology

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2002

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Vol. 8, nr 1

63-67

EN

A transfer matrix approach has been worked out to test the predictions of the three-step molecular-field renormalization group (MFRG) for square Ising clusters with linear size up t o L + 11. The convergence of the finite-size critical couplings and the critical exponents towards the exact values is shown.

7

Methods of computer simulations of phase transitions in the Ashkin - Teller model

Musiał G., Dębski L., Kamieniarz G., Rogiers J.

Computational Methods in Science and Technology

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2001

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Vol. 7, nr 1

67-82

EN

The complementary Monte Carlo and series expansions methods of computer simulations have been described to investigate the critical behaviour of the Ashkin-Teller model in three dimensions. In the first method the invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region has been exploited and some critical points on the phase boundaries have been calculated in the regions where the continuous transitions are expected. The continuity of the order parameter on the critical lines is verified by a finite size scaling analysis. Large-scale simulations have been performed on SGI Power Challenge XL and L supercomputers using the 64-bit random number generator. The numerically generated series expansions method is described for which the effective algorithm for generation of graphs starting from polygons and based on collapsing the unlinked vertices, is introduced. The new feature of our algorithm is that for each graph we introduce new links between unlinked vertices and we decorate bonds with new vertices, so that more complex graphs in an early stage of the graph generation procedure are obtained. The resulting set of graphs enables the application of the series expansions method and achievement of the precision of allocation of points on the phase diagram comparable to the precision of the Monte Carlo method, i.e. at least 3 decimal digits.

8

Z badań nad dziejami ekonomii jako dziedziny nauki. W 210-tą rocznicę śmierci Adama Smitha (1723-1790)

Musiał G.

Kwartalnik Historii Nauki i Techniki

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2000

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R. 45, nr 1

7--30

PL

W lipcu 2000 r. minie 210 lat od śmierci Adama Smitha. Jego spuścizna naukowa jest przejawem połączenia uzdolnień filozofa, ekonomisty, męża stanu. Smith wytyczył drogi ekonomicznego myślenia, opanował dostępny w jego czasach materiał empiryczny, przeniknął znaczenie ówczesnych wydarzeń społecznych i dążył do wszechstronnego ich wyjaśnienia. Główne dzieło Smitha Badania nad naturą i przyczynami bogactwa narodów, które ukazało się w 1776 r. jest nadal pozycją znaczącą, choć ekonomia dawno przekroczyła perspektywę Smithowskiej dynamiki i przybrała inną postać. Warsztat badawczy Smitha cechuje rzetelna wiedza ekonomiczna, wirtuozeria analityczna i kultura myślenia. Potrzeba wspomnienia tego uczonego wynika stąd, że jego dzieło zachowuje swoją ważność dla ekonomii i historii nauki, usiłujących zrozumieć mechanizmy rozwojowe nauki. Dzieło to zamyka jeden okres rozwojowy ekonomii z właściwym jej obrazem świata oraz metodą jego oglądu i rozpoczyna nowy okres, dla którego podejście Smithowskie stało się płaszczyzną odniesienia. Główne idee tego czy innego myśliciela można rozpatrywać z rożnego punktu widzenia w zależności od przyjętego celu dociekań. I tak, podejście genetyczne pozwala na ujawnienie ewolucji od najwcześniejszych, prostych i często odmiennychsformułowań w jakimś młodzieńczym dziele przez rożne dzieła przejściowe. Idee danego uczonego można też postrzegać jako odzwierciedlenie szerokich nurtów myśli i ówczesnych kontrowersji. Wreszcie można odnosić się w pracy badawczej do idei i poglądów danego uczonego zawartych w jakimś szczególnym dziele, w którym unaoczniło się jego stanowisko w najbardziej dojrzałej formie.Wszystkie podejścia są uprawnione, choć każde z nich ma swoje ograniczenia. W niniejszym artykule znalazły zastosowanie pierwsze i trzecie podejście, z tym, że dominuje to ostatnie. Nie sprowadza się ono jednak do rekonstrukcji całokształtu poglądów Adama Smitha we wspomnianym już głównym jego dziele Badania nad naturą i przyczynami bogactwa narodów, ani też do sprawozdania z przeprowadzonej interpretacji jego poglądów. Jeśli studiowanie dorobku dawnych uczonych ma przynieść korzyść naukom społecznym, w tym ekonomii, to powinno mieć ono jakiś związek z zainteresowaniami teoretyków i społeczeństwa naszych czasów. Ekonomiści od ponad dwóch stuleci znajdują się pod wpływem hipotezy Adama Smitha o „niewidzialnej ręce”, zaś w ostatniej dekadzie zainteresowanie tą hipotezą przejawiają politycy, a przede wszystkim etycy, specjalizujący się w biznesie. Z hipotezą tą kojarzą oni wszyscy dobroczynne działanie rynku, wolność gospodarczą jednostki nie tkniętej przez wpływy społeczne, całkowitą atomizację społeczeństwa. W artykule tym zajęłam się bliżej ową hipotezą. Nie ma potrzeby podkreślać jej znaczenia. Niewiele, jeśli w ogóle, jest w ekonomii problemów,które prędzej czy później, nie wywołują pytania, czym jest wolność gospodarcza.

EN

Adam Smith was one of the representatives of the Scottish Enlightenment. Until recently, he was known as the author of two books: Theory of Moral Sentiments, which was published in 1759, and Inquiry into the Nature and Causes of the Wealth of Nations published seventeen years later. The latter book is still of great importance even though economy has long since gone beyond the perspective of Smith’s dynamics and assumed a different form. The need to remember the talented scholar, who showed great analytical virtuosity, results from the fact that his work still maintains its significance for both economy and the history of science, as they attempt to understand the developmental mechanisms o f science. The article presents the origin of Adam Smith’s views and explains the metaphor of the ’’invisible hand”, which is in extricably associated with Smith’s name. At the end of the article, the author expresses her attitude to some contemporary scholars’ interpretations of the ’’invisible hand” metaphor (F. Braudel, N. Chomsky).

9

The algorithm for generation of connected closed linear graphs for lattice - spin systems

Musiał G.

Computational Methods in Science and Technology

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1998

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Vol. 4

75-78

EN

The paper presents the algorithm and subroutine GRGEN for generation of connected closed linear graphs utilized in perturbation expansions. Starting with polygons and subsequently choosing pairs of not directly connected vertices in already generated graphs, the algorithm consequently projects one vertex onto the other one in the pair, generating only topologically different graphs. This algorithm is independent of the choice of a specific model and lattice geometry. The procedure is written in the FORTRAN 77 language, and is available at the Poznań Supercomputing and Networking Center on the SGI Power Challenge XL supercomputer.