Search results for: curvature entropy

Authors: Hao Wang, Shengchun Wang, Weidong Wang

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On the influence of train vibration and environmental noise on the measurement of track wear, we proposed a method for automatic extraction of circular arc on the inner or outer side of the rail waist and achieved the high-precision registration of rail profile. Firstly, a polynomial fitting method based on truncated residual histogram was proposed to find the optimal fitting curve of the profile and reduce the influence of noise on profile curve fitting. Then, based on the curvature distribution characteristics of the fitting curve, the interval search algorithm based on dynamic window’s maximum curvature entropy was proposed to realize the automatic segmentation of small circular arc. At last, we fit two circle centers as matching reference points based on small circular arcs on both sides and realized the alignment from the measured profile to the standard designed profile. The static experimental results show that the mean and standard deviation of the method are controlled within 0.01mm with small measurement errors and high repeatability. The dynamic test also verified the repeatability of the method in the train-running environment, and the dynamic measurement deviation of rail wear is within 0.2mm with high repeatability.

Keywords: curvature entropy, profile registration, rail wear, structured light, train-running

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Authors: Hamidreza Fazlollahi

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The Re ́nyi entropy comprises a group of data estimates that sums up the well-known Shannon entropy, acquiring a considerable lot of its properties. It appears as unqualified and restrictive entropy, relative entropy, or common data, and has found numerous applications in information theory. In the Re ́nyi’s argument, the area law of the black hole entropy plays a significant role. However, the total entropy can be modified by some quantum effects, motivated by the randomness of a system. In this note, by employing this modified entropy relation, we have derived corrections to Friedmann equations. Taking this entropy associated with the apparent horizon of the Friedmann-Robertson-Walker Universe and assuming the first law of thermodynamics, dE=T_A (dS)_A+WdV, satisfies the apparent horizon, we have reconsidered expanding Universe. Also, the second thermodynamics law has been examined.

Keywords: Friedmann equations, dark energy, first law of thermodynamics, Reyni entropy

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Authors: Petr Jizba, Jan Korbel

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We combine an axiomatics of Rényi with the q-deformed version of Khinchin axioms to obtain a measure of information (i.e., entropy) which accounts both for systems with embedded self-similarity and non-extensivity. We show that the entropy thus obtained is uniquely solved in terms of a one-parameter family of information measures. The ensuing maximal-entropy distribution is phrased in terms of a special function known as the Lambert W-function. We analyze the corresponding ‘high’ and ‘low-temperature’ asymptotics and reveal a non-trivial structure of the parameter space.

Keywords: multifractals, Rényi information entropy, THC entropy, MaxEnt, heavy-tailed distributions

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Authors: I. Arockiarani

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The focus of the paper is to furnish the entropy measure for a neutrosophic set and neutrosophic soft set which is a measure of uncertainty and it permeates discourse and system. Various characterization of entropy measures are derived. Further we exemplify this concept by applying entropy in various real time decision making problems.

Keywords: entropy measure, Hausdorff distance, neutrosophic set, soft set

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Authors: M. Yushalify Misro, Ahmad Ramli, Jamaludin M. Ali

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Bezier curves have useful properties for path generation problem, for instance, it can generate the reference trajectory for vehicles to satisfy the path constraints. Both algorithms join cubic Bezier curve segment smoothly to generate the path. Some of the useful properties of Bezier are curvature. In mathematics, the curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line. Another extrinsic example of curvature is a circle, where the curvature is equal to the reciprocal of its radius at any point on the circle. The smaller the radius, the higher the curvature thus the vehicle needs to bend sharply. In this study, we use Bezier curve to fit highway-like curve. We use the different approach to finding the best approximation for the curve so that it will resemble highway-like curve. We compute curvature value by analytical differentiation of the Bezier Curve. We will then compute the maximum speed for driving using the curvature information obtained. Our research works on some assumptions; first the Bezier curve estimates the real shape of the curve which can be verified visually. Even, though, the fitting process of Bezier curve does not interpolate exactly on the curve of interest, we believe that the estimation of speed is acceptable. We verified our result with the manual calculation of the curvature from the map.

Keywords: speed estimation, path constraints, reference trajectory, Bezier curve

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Authors: Graziano Chesi

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The topological entropy plays a key role in linear dynamical systems, allowing one to establish the existence of stabilizing feedback controllers for linear systems in the presence of communications constraints. This paper addresses the determination of a robust value of the topological entropy in nonlinear dynamical systems, specifically the largest value of the topological entropy over all linearized models in a region of interest of the state space. It is shown that a sufficient condition for establishing upper bounds of the sought robust value of the topological entropy can be given in terms of a semidefinite program (SDP), which belongs to the class of convex optimization problems.

Keywords: non-linear system, communication constraint, topological entropy

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Authors: M. Yaqub Khan, Usman Shabbir

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History of plasma reveals that continuous struggle of experimentalists and theorists are not fruitful for confinement up to now. It needs a change to bring the research through entropy. Approximately, all the quantities like number density, temperature, electrostatic potential, etc. are connected to entropy. Therefore, it is better to change the way of research. In ion temperature gradient mode with the help of Braginskii model, Boltzmannian electrons, effect of velocity shear is studied inculcating entropy in the magnetoplasma. New dispersion relation is derived for ion temperature gradient mode, and dependence on entropy gradient drift is seen. It is also seen velocity shear enhances the instability but in anomalous transport, its role is not seen significantly but entropy. This work will be helpful to the next step of tokamak and space plasmas.

Keywords: entropy, velocity shear, ion temperature gradient mode, drift

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Authors: Gary Miller, Andriy Didenko, David Allison

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The boundary of a region in the plane shrinks according to its curvature. A simple algorithm based upon this motion by curvature performed by a spreadsheet simulates the evaporation/dissipation behavior of oil spill boundaries.

Keywords: mathematical modeling, oil, evaporation, dissipation, boundary

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Authors: Arvinder Kaur, Deepti Chopra

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Software Entropy Metrics for bug prediction have been validated on various software systems by different researchers. In our previous research, we have validated that Software Entropy Metrics calculated for Mozilla subsystem’s predict the future bugs reasonably well. In this study, the Software Entropy metrics are calculated for a subsystem of Android and it is noticed that these metrics are not suitable for bug prediction. The results are compared with a subsystem of Mozilla and a comparison is made between the two software systems to determine the reasons why Software Entropy metrics are not applicable for Android.

Keywords: android, bug prediction, mining software repositories, software entropy

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Authors: Subha D. Puthankattil, Paul K. Joseph

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Analyzing brain signals of the patients suffering from the state of depression may lead to interesting observations in the signal parameters that is quite different from a normal control. The present study adopts two different methods: Time frequency domain and nonlinear method for the analysis of EEG signals acquired from depression patients and age and sex matched normal controls. The time frequency domain analysis is realized using wavelet entropy and approximate entropy is employed for the nonlinear method of analysis. The ability of the signal processing technique and the nonlinear method in differentiating the physiological aspects of the brain state are revealed using Wavelet entropy and Approximate entropy.

Keywords: EEG, depression, wavelet entropy, approximate entropy, relative wavelet energy, multiresolution decomposition

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Authors: Ahmadali Shirazytabar, Hamidreza Namazi

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The inherent irreversibility of thermo-acoustics primarily in the stack region causes poor efficiency of thermo-acoustic engines which is the major weakness of these devices. In view of the above, this study examines entropy generation in the stack of a thermo-acoustic system. For this purpose two parallel plates representative of the stack is considered. A general equation for entropy generation is derived based on the Second Law of thermodynamics. Assumptions such as Rott’s linear thermo-acoustic approximation, boundary layer type flow, etc. are made to simplify the governing continuity, momentum and energy equations to achieve analytical solutions for velocity and temperature. The entropy generation equation is also simplified based on the same assumptions and then is converted to dimensionless form by using characteristic entropy generation. A time averaged entropy generation rate followed by a global entropy generation rate are calculated and graphically represented for further analysis and inspecting the effect of different parameters on the entropy generation.

Keywords: thermo-acoustics, entropy, second law of thermodynamics, Rott’s linear thermo-acoustic approximation

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Authors: Muhammad Sheraz, Vasile Preda, Silvia Dedu

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Concepts of econophysics are usually used to solve problems related to uncertainty and nonlinear dynamics. In the theory of option pricing the risk neutral probabilities play very important role. The application of entropy in finance can be regarded as the extension of both information entropy and the probability entropy. It can be an important tool in various financial methods such as measure of risk, portfolio selection, option pricing and asset pricing. Gulko applied Entropy Pricing Theory (EPT) for pricing stock options and introduced an alternative framework of Black-Scholes model for pricing European stock option. In this article, we present solutions to maximum entropy problems based on Tsallis, Weighted-Tsallis, Kaniadakis, Weighted-Kaniadakies entropies, to obtain risk-neutral densities. We have also obtained the value of European call and put in this framework.

Keywords: option pricing, Black-Scholes model, Tsallis entropy, Kaniadakis entropy, weighted entropy, risk-neutral density

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Authors: Y. S. Oh, P. Abhishesh, B. S. Ryuh

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This paper presents the method of trajectory planning by the robot end-effector which accounts for more accurate and smooth differential geometry of the ruled surface generated by tool line fixed with end-effector based on the methods of curvature theory of ruled surface and the dual curvature theory, and focuses on the underlying relation to unite them for enhancing the efficiency for trajectory planning. Robot motion can be represented as motion properties of the ruled surface generated by trajectory of the Tool Center Point (TCP). The linear and angular properties of the six degree-of-freedom motion of end-effector are computed using the explicit formulas and functions from curvature theory and dual curvature theory. This paper explains the complete dualization of ruled surface and shows that the linear and angular motion applied using the method of dual curvature theory is more accurate and less complex.

Keywords: dual curvature theory, robot end effector, ruled surface, TCP (Tool Center Point)

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Authors: Septimia Sarbu

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The entropy rate of a stochastic process is a fundamental concept in information theory. It provides a limit to the amount of information that can be transmitted reliably over a communication channel, as stated by Shannon's coding theorems. Recently, researchers have focused on developing new measures of information that generalize Shannon's classical theory. The aim is to design more efficient information encoding and transmission schemes. This paper continues the study of generalized entropy rates, by deriving a closed-form solution to the Sharma-Mittal entropy rate for Gaussian processes. Using the squeeze theorem, we solve the limit in the definition of the entropy rate, for different values of alpha and beta, which are the parameters of the Sharma-Mittal entropy. In the end, we compare it with Shannon and Rényi's entropy rates for Gaussian processes.

Keywords: generalized entropies, Sharma-Mittal entropy rate, Gaussian processes, eigenvalues of the covariance matrix, squeeze theorem

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Authors: Darrol Stanley, Levan Efremidze, Jannie Rossouw

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We investigate the predictability of the USD/ZAR (South African Rand) exchange rate with sample entropy analytics for the period of 2004-2015. We calculate sample entropy based on the daily data of the exchange rate and conduct empirical implementation of several market timing rules based on these entropy signals. The dynamic investment portfolio based on entropy signals produces better risk adjusted performance than a buy and hold strategy. The returns are estimated on the portfolio values in U.S. dollars. These results are preliminary and do not yet account for reasonable transactions costs, although these are very small in currency markets.

Keywords: currency trading, entropy, market timing, risk factor model

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Authors: Hyunwoo Song, Jeong-Min Lee, Yongseok Kim, Junghan Yun, Jungin Byun, Jae-Mean Koo, Chang-Sung Seok

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To avoid the damage of gas turbine blade from high-temperature, thermal barrier coating (TBC) is applied on the blade. However, it is damaged by thermal fatigue during the operation of gas turbine, and this damage lead to delamination of TBC between top coat and bond coat. The blade can be damaged after the failure of TBC, so durability evaluation of TBC should be performed. The durability of thermal barrier coating was decreased according to the increase of temperature, because thermal stress according to increase of temperature. Also, the curvature can be affect to durability of TBC, because the stress is determined by the shape of the TBC. Therefore, the effect of temperature and curvature on the stress should be evaluated. In this study, finite element analysis according to temperature and curvature were performed in the same condition of Kim et al. Finally, the stress was evaluated from the finite element analysis results according to temperature and curvature.

Keywords: curvature, finite element analysis, thermal barrier coating, thermal fatigue, temperature

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Authors: Fahad Alsharari, Mohd Salmi Md. Noorani

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A Motzkin shift is a mathematical model for constraints on genetic sequences. In terms of the theory of symbolic dynamics, the Motzkin shift is nonsofic, and therefore, we cannot use the Perron-Frobenius theory to calculate its topological entropy. The Motzkin shift M(M,N) which comes from language theory, is defined to be the shift system over an alphabet A that consists of N negative symbols, N positive symbols and M neutral symbols. For an x in the full shift AZ, x is in M(M,N) if and only if every finite block appearing in x has a non-zero reduced form. Therefore, the constraint for x cannot be bounded in length. K. Inoue has shown that the entropy of the Motzkin shift M(M,N) is log(M + N + 1). In this paper, we find a new method of calculating the topological entropy of the Motzkin shift M(M,N) without any measure theoretical discussion.

Keywords: entropy, Motzkin shift, mathematical model, theory

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Authors: E. Dehghan, R. Dehghan

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Generally, the slender structural walls have flexural behavior. Since behavior of bending members can be explained by moment–curvature relation, therefore, an analytical model is proposed based on moment–curvature relation for slender structural walls. The moment–curvature relationships of RC sections are constructed through section analysis. Governing equations describing the bond-slip behavior in walls are derived and applied to moment–curvature relations. For the purpose of removing the imprecision in analytical results, the plastic hinge length is included in the finite element modeling. Finally, correlation studies between analytical and experimental results are conducted with the objective to establish the validity of the proposed algorithms. The results show that bond-slip effect is more significant in walls subjected to larger axial compression load. Moreover, preferable results are obtained when ultimate strain of concrete is assumed conservatively.

Keywords: nonlinear analysis, slender structural walls, moment-curvature relation, bond-slip, plastic hinge length

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Authors: Asadollah Bahrami

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In the present study, spectral radiative entropy generation is analyzed in a furnace filled with a mixture of H₂O, CO₂ and soot at radiative equilibrium. For the angular and spatial discretization of the radiative transfer equation and radiative entropy generation equations, the discrete ordinates method and the finite volume method are used, respectively. Spectral radiative properties are obtained using the correlated-k (CK) non-gray model with updated parameters based on the HITEMP2010 high-resolution database. In order to evaluate the effects of the location of the heat source, boundary condition and wall emissivity on radiative entropy generation, five cases are considered with different conditions. The spectral and total radiative entropy generation in the system are calculated for all cases and the effects of mentioned parameters on radiative entropy generation are attentively analyzed and finally, the optimum condition is especially presented. The most important results can be stated as follows: Results demonstrate that the wall emissivity has a considerable effect on the radiative entropy generation. Also, irreversible radiative transfer at the wall with lower temperatures is the main source of radiative entropy generation in the furnaces. In addition, the effect of the location of the heat source on total radiative entropy generation is less than other factors. Eventually, it can be said that characterizing the effective parameters of radiative entropy generation provides an approach to minimizing the radiative entropy generation and enhancing the furnace's performance practicality.

Keywords: spectral radiative entropy generation, non-gray medium, correlated k(CK) model, heat source

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Authors: Sheema Shuja Khattak, Gule Saman, Imran Khan, Abdus Salam

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Image segmentation plays an important role in medical imaging applications. Therefore, accurate methods are needed for the successful segmentation of medical images for diagnosis and detection of various diseases. In this paper, we have used maximum entropy to achieve image segmentation. Maximum entropy has been calculated using Shannon, Renyi, and Tsallis entropies. This work has novelty based on the detection of skin lesion caused by the bite of a parasite called Sand Fly causing the disease is called Cutaneous Leishmaniasis.

Keywords: shannon, maximum entropy, Renyi, Tsallis entropy

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Authors: M. Alipanah, A. Ranjbar, E. Farnad, F. Alipanah

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Entropy generation of an Al2O3-water nanofluid due to heat transfer and fluid friction irreversibility has been investigated in a square cavity subject to different side wall temperatures using a nanofluid for natural convection flow. This study has been carried out for the pertinent parameters in the following ranges: Rayleigh number between 104 to 107 and volume fraction between 0 to 0.05. Based on the obtained dimensionless velocity and temperature values, the distributions of local entropy generation, average entropy generation and average Bejan number are determined. The results are compared for a pure fluid and a nanofluid. It is totally found that the heat transfer and entropy generation of the nanofluid is more than the pure fluid and minimum entropy generation and Nusselt number occur in the pure fluid at any Rayleigh number. Results depict that the addition of nanoparticles to the pure fluid has more effect on the entropy generation as the Rayleigh number goes up.

Keywords: entropy generation, natural convection, bejan number, nuselt number, nanofluid

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Authors: Abul Kayum Zarzis Alam

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Death is not a terminal; it is just a junction. From Agamas to Vedas, from Buddhism to Judaism, all the major scriptures and religions of the world always do converge to this hypothesis of death. Death is the ultimate catastrophe of life and it is the genesis of every religion on this Earth. Several hundred thousand years ago, the Homo Sapiens in Paleolithic age introduced the notion of religion on this Earth in its most primitive form just to escape from death and natural catastrophes through their belief in supernatural things which created the sense of superstition among the Homo Sapiens which has only increased over time. This sense of superstition and belief in supernatural things are building blocks of religion. Religion is like entropy, a degree of disorder. Entropy for an irreversible system like our own Universe always increases. Same is happening to our human civilization where the disorder had been increasing over time. The degree of this disorder of human civilization is religion divides and conquers over the human civilization of Earth. Religion is the human entropy which had been governing and will govern us. Just like entropy, religion is also an essential intrinsic property of the system which makes the system evolved. We have to optimize this ambivalence of the human entropy to make our civilization an inclusive and sustainable one.

Keywords: death, earth, entropy, Homo sapiens, religion and human entropy

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Authors: Stoyan Nedeltchev, Markus Schubert

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The gas holdup fluctuations in a bubble column (0.15 m in ID) have been recorded by means of a conductivity wire-mesh sensor in order to extract information about the main transition velocities. These parameters are very important for bubble column design, operation and scale-up. For this purpose, the classical definition of the Shannon entropy was modified and used to identify both the onset (at UG=0.034 m/s) of the transition flow regime and the beginning (at UG=0.089 m/s) of the churn-turbulent flow regime. The results were compared with the Kolmogorov entropy (KE) results. A slight discrepancy was found, namely the transition velocities identified by means of the KE were shifted to somewhat higher (0.045 and 0.101 m/s) superficial gas velocities UG.

Keywords: bubble column, gas holdup fluctuations, modified Shannon entropy, Kolmogorov entropy

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Authors: Stoyan Nedeltchev, Markus Schubert

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By means of the ultrafast X-ray tomography facility, data were obtained at different superficial gas velocities U_G in a bubble column (0.1 m in ID) operated with an air-deionized water system at ambient conditions. Raw reconstructed images were treated by both the information entropy (IE) and the reconstruction entropy (RE) algorithms in order to identify the main transition velocities in a bubble column. The IE values exhibited two well-pronounced minima at U_G=0.025 m/s and U_G=0.085 m/s identifying the boundaries of the homogeneous, transition and heterogeneous regimes. The RE extracted from the central region of the column’s cross-section exhibited only one characteristic peak at U_G=0.03 m/s, which was attributed to the transition from the homogeneous to the heterogeneous flow regime. This result implies that the transition regime is non-existent in the core of the column.

Keywords: bubble column, ultrafast X-ray tomography, information entropy, reconstruction entropy

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Authors: Jason Clarke, Michaela Porubanova, Angela Mazzoli, Gulsah Kut

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What accounts for our perception that time inexorably passes in one direction, from the past to the future, the so-called arrow of time, given that the laws of physics permit motion in one temporal direction to also happen in the reverse temporal direction? Modern physics says that the reason for time’s unidirectional physical arrow is the relationship between time and entropy, the degree of disorder in the universe, which is evolving from low entropy (high order; thermal disequilibrium) toward high entropy (high disorder; thermal equilibrium), the second law of thermodynamics. Accordingly, our perception of the direction of time, from past to future, is believed to emanate as a result of the natural evolution of entropy from low to high, with low entropy defining our notion of ‘before’ and high entropy defining our notion of ‘after’. Here we explored this proposed relationship between entropy and the perception of time’s arrow. We predicted that if the brain has some mechanism for detecting entropy, whose output feeds into processes involved in constructing our perception of the direction of time, presentation of violations to the expectation that low entropy defines ‘before’ and high entropy defines ‘after’ would alert this mechanism, leading to measurable behavioral effects, namely a disruption in duration perception. To test this hypothesis, participants were shown briefly-presented (1000 ms or 500 ms) computer-generated visual dynamic events: novel 3D shapes that were seen either to evolve from whole figures into parts (low to high entropy condition) or were seen in the reverse direction: parts that coalesced into whole figures (high to low entropy condition). On each trial, participants were instructed to reproduce the duration of their visual experience of the stimulus by pressing and releasing the space bar. To ensure that attention was being deployed to the stimuli, a secondary task was to report the direction of the visual event (forward or reverse motion). Participants completed 60 trials. As predicted, we found that duration reproduction was significantly longer for the high to low entropy condition compared to the low to high entropy condition (p=.03). This preliminary data suggests the presence of a neural mechanism that detects entropy, which is used by other processes to construct our perception of the direction of time or time’s arrow.

Keywords: time perception, entropy, temporal illusions, duration perception

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Authors: Shi-hong Lu, Chao-xun Liu, Yi-feng Zhu

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Ultrasonic shot peening (USP) on AALY12 sheet was studied. Several parameters (arc heights, surface roughness, surface topography and microhardness) with different USP process parameters were measured. The research proposes that the radius of curvature of shot peened sheet increases with time and electric current decreasing, while it increases with pin diameter increasing, and radius of curvature reaches a saturation level after a specific processing time and electric current. An empirical model of the relationship between radius of curvature and pin diameter, electric current, time was also obtained. The research shows that the increment of surface and vertical microhardness of material is more obvious with longer time and higher value of electric current, which can be up to 20% and 28% respectively.

Keywords: USP forming, surface properties, radius of curvature, residual stress

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Authors: Amel Tayari, Atef Eljerry, Mourad Magherbi

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The paper reports a numerical investigation of the entropy generation analysis due to mixed convection in laminar flow through a channel filled with porous media. The second law of thermodynamics is applied to investigate the entropy generation rate. The Darcy-Brinkman Model is employed. The entropy generation due to heat transfer and friction dissipations has been determined in mixed convection by solving numerically the continuity, momentum and energy equations, using a control volume finite element method. The effects of Darcy number, modified Brinkman number and the Rayleigh number on averaged entropy generation and averaged Nusselt number are investigated. The Rayleigh number varied between 103 ≤ Ra ≤ 105 and the modified Brinkman number ranges between 10-5 ≤ Br≤ 10-1 with fixed values of porosity and Reynolds number at 0.5 and 10 respectively. The Darcy number varied between 10-6 ≤ Da ≤10.

Keywords: entropy generation, porous media, heat transfer, mixed convection, numerical methods, darcy, brinkman

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Authors: J. Juan Peña, J. Morales, J. García-Ravelo, J. García-Martínes

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It is known that by introducing alternative forms of exponential and logarithmic functions, the Tsallis entropy Sq is itself a generalization of Shannon entropy S. In this work, from a deformation through a scaling function applied to the differential operator, it is possible to generate a q-deformed calculus as well as a q-deformed arithmetic, which not only allows generalizing the exponential and logarithmic functions but also any other standard function. The updated q-deformed differential operator leads to an updated integral operator under which the functions are integrated together with a weight function. For each differentiable function, it is possible to identify its q-deformed partner, which is useful to generalize other algebraic relations proper of the original functions. As an application of this proposal, in this work, a generalization of exponential and logarithmic functions is studied in such a way that their relationship with the thermodynamic functions, particularly the entropy, allows us to have a q-deformed expression of these. As a result, from a particular scaling function applied to the differential operator, a q-deformed arithmetic is obtained, leading to the generalization of the Tsallis entropy.

Keywords: q-calculus, q-deformed arithmetic, entropy, exponential functions, thermodynamic functions

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Authors: Mohammad A. U. Khan, Omar A. Kittaneh, M. Akbar, Tariq M. Khan, Husam A. Bayoud

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The Shannon Entropy measure has been widely used for measuring uncertainty. However, in partial settings, the histogram is used to estimate the underlying distribution. The histogram is dependent on the number of bins used. In this paper, a modification is proposed that makes the Shannon entropy based on histogram consistent. For providing the benefits, two application are picked in medical image processing applications. The simulations are carried out to show the superiority of this modified measure for image segmentation problem. The improvement may be contributed to robustness shown to uneven background in images.

Keywords: Shannon entropy, medical image processing, image segmentation, modification

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Authors: Yuriy A. Shevchenko, Konstantin V. Nefedev

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We consider the antiferromagnetic systems of Ising spins located at the sites of the hexagonal, triangular and pyrochlore lattices. Such systems can be diluted to a certain concentration level by randomly replacing the magnetic spins with nonmagnetic ones. Quite recently we studied density of states (DOS) was calculated by the Wang-Landau method. Based on the obtained data, we calculated the dependence of the residual entropy (entropy at a temperature tending to zero) on the dilution concentration for quite large systems (more than 2000 spins). In the current study, we obtained the same data for small systems (less than 20 spins) by a complete search of all possible magnetic configurations and compared the result with the result for large systems. The shape of the curve remains unchanged in both cases, but the specific values of the residual entropy are different because of the finite size effect.

Keywords: entropy, pyrochlore, spin ice, Wang-Landau algorithm

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