PHYSICAL REVIEW E VOLUME 61, NUMBER 4 APRIL 2000 Stochastic model of hysteresis L. PaŽl KFKI Atomic Energy Research Institute, P.O. Box 49, 1525 Budapest, Hungary Received 20 May 1999 The methods of the probability theory have been used in order to build up a model of hysteresis which is different from the well-known Preisach model. It is assumed that the system consists of large number of abstract particles in which the variation of an external control parameter e.g., the magnetic field may result in transitions between two states S ( ) and S ( ). The state of a particle is characterized by the value 1 or 1 of a random variable e.g., the magnetization direction parallel or antiparallel to the magnetic field . The transitions are governed by two further random variables corresponding to the S ( ) S ( ) and the S ( ) S ( ) transitions e.g., ``up switching'' and ``down switching magnetic field'' . The method presented here makes it possible to calculate the probability distribution and consequently the expectation value of the number of particles in the S ( ) or S ( )) state for both increasing and decreasing parameter values, i.e., the hysteresis curves of the transitions can be determined. It turns out that the reversal points of the control parameter are Markov points which determine the stochastic evolution of the process. It has been shown that the branches of the hysteresis loop are converging to fixed limit curves when the number of cyclic back-and- forth variations of the control parameter between two consecutive reversal points is large enough. This con- vergence to limit curves gives a clear explanation of the accommodation process. The accommodated minor loops show the return-point memory property but this property is obviously absent in the case of nonaccom- modated minor loops which are not congruent and generally not closed. In contrast to the traditional Preisach model the reversal point susceptibilities are nonzero finite values. The stochastic model can provide a surpris- ingly good approximation of the Raylaigh quadratic law when the external parameter varies between two sufficiently small values. The practical benefits of the model can be seen in the numerical analysis of the derived equations. On one hand the calculated curves are in good qualitative agreement with the experimental observations and on the other hand, the estimation of the joint distribution function of the up and down switching fields can be performed by using the measured hysteresis curves. PACS number s : 02.50.Ga, 02.50.Ey, 75.60.Ej I. INTRODUCTION II. DESCRIPTION OF THE MODEL Let us assume that the unit volume of the system consists The phenomenon of the hysteresis, understood in a gen- of many small abstract regions, called ``particles'' which are eral sense, has been investigated so intensively for many characterized by four random variables , , d , and u . decades that any list of references would be far from com- The absolute value of the particle magnetization is denoted plete by any standards. It is very fortunate that in the last few by . If the particle magnetization is parallel antiparallel to years outstanding monographs 1­4 have been published in the external magnetic field H then the particle is in the state this field, and thus the author does not feel obliged to cite the S ( )(S ( )) and 1( 1). The random value d corre- large amount of old but important references. However, it is sponds to a local field at which the state S ( ) jumps to the considered important to mention two papers of KaŽdaŽr 5,6 state S ( ) and similarly the S ( ) S ( ) transition occurs at whose work has played a stimulating role in getting to the the random local field u . For simplicity the u and d will idea of reconsideration of the hysteresis theory by the present be called the U and D fields. These two random variables author. There is no doubt that the abstract reformulation of express the obvious fact that each particle ``feels'' not only the Preisach model 7 given by Krasnoselskii and Pok- the external magnetic field H, but also the interaction field rovskii 8 and summarized by Mayergoyz 1 in his book due to the adjacent particles and the random field originated from the inhomogenities of the surrounding environment. resulted in an improved mathematical clarity in the hyster- These particles characterized by the random variables esis theory, but the stochastic nature of the hysteresis still has , , not been treated with sufficient mathematical rigor 16 . d , and u can be regarded as ``independent'' abstract elements of the system, and they will be called ``hysterons.'' The aim of the present paper is to define a stochastic Figure 1 illustrates a possible realization of transitions model of hysteresis and to derive exact equations for the S ( ) S ( ) of a hysteron. The transition curves form a ran- probability distribution functions describing the state varia- dom rectangular hysteresis loop which is almost in all cases tions in hysteretic processes as a function of increasing as asymmetrical in the coordinate system of magnetization ver- well as decreasing control parameters. The vocabulary of sus external magnetic field since the U and D fields are sup- magnetic hysteresis will be used for convenience from now posed to be random. on, however, the concepts can easily be generalized for any Let us denote the hysterons in a system by hysteretic phenomenon. h1 ,h2 , . . . ,hN , and let N ( ) be the set of indices and n( ) 1063-651X/2000/61 4 /3490 11 /$15.00 PRE 61 3490 © 2000 The American Physical Society PRE 61 STOCHASTIC MODEL OF HYSTERESIS 3491 N H, n( ) pn( ) H n( ) 0 N N p n( ) n( ) H , n( ) 0 4 we need the probability of finding n( ) hysterons in the state S ( ) at the external field H which is the endpoint of a well- defined magnetization prehistory. The determination of this probability and the derivation of the equations for ``up'' and ``down'' magnetizations versus magnetic field will be the task of the next section. III. DERIVATION OF THE FUNDAMENTAL EQUATIONS FIG. 1. A possible realization of the transition S ( ) S ( ). A. Some basic relations the number of hysterons which are in the state S ( ) at a Let us denote by H(x,y C) the joint distribution function given external field H. In this case the magnetization of the of the random U and D fields. From the physical point of system is given by the stochastic equation view it is quite obvious that the U field cannot be smaller than the D field, so the stochastic inequality u d must be N satisfied. It is easy to show 9 that the joint distribution function of n( ) k k , u and d satisfying the condition C u d k 1 can be written in the form where P u x, d y u d k is the absolute value of the magnetization of the hysteron hk , while H x,y C x y dx h x ,y x y dy k 1, if k N( ), 1, if k N ( ). , 5 x dx h x ,y dy Since the random variables 1 , 2 , . . . , N are mutually in- dependent and have the same probability distribution func- tion where (x) is the unit step function. It is clear that the joint density function of the U and D fields can be given by P k x L x , k 1,2, . . . ,N, h x,y it is obvious that the characteristic function of the distribu- h x,y C x y , 6 x tion function dx h x ,y dy P n( ) x Rn( ) x 1 provided that the condition C is valid. can be written in the form We need in the sequel two conditional probability distri- bution functions n( ) P n( ) ei xdRn( ) x N , u x C H x, C Fu x C 2 x x dx h x ,y dy where x dx h x ,y dy ei xdL x ei xdL x . 3 0 7 In order to calculate the characteristic function and 3492 L. PAŽL PRE 61 P d y C H ,y C Fd y C y dy h x ,y dx y . 8 dy h x ,y dx y Evidently Fu(x C) is the probability that the U field of a given hysteron is not larger than x, while Fd(y C) is the probability that the D field is not larger than y assuming in both cases that the condition C is fulfilled. By using the Di- richlet's theorem for changing the sequence of integration it is obvious that FIG. 2. Two equivalent H(t) curves. x dx h x ,y dy dy h x ,y dx . of extrema is the same for both curves, but the time distance y and the shape of sections between the consecutive extrema For the sake of further considerations it is necesarry to are different. introduce two transition probabilities denoted by w It is assumed that the magnetizing process which consists u(Hl H) and w of random transitions S ( ) S ( ) of hysterons does not d(Hu H). Let Hl be a fixed value of the external mag- netic field and let us suppose that the state of a given hys- ``feel'' the variation speed of H(t) between the consecutive teron is S ( ) at H extremum values, i.e., the magnetizing process is static. The l . By using elementary theorems, it can be proved that if the external field increases monotonically from evolution of the process in each subinterval Hj ,Hj 1 , j H 1,2, . . . , is stochasically determined by the extremum H l to H Hl then j and by the actual values of H(t) following Hj , but the pro- H x cess does not depend on the time derivative of H(t). This dx h x,y dy property is called rate independence in the nonstochastic H w l u Hl H 9 theory of hysteresis 2 but it will be applied in this stochas- x dx h x,y dy tic theory too. Denote the maximum reversal fields by odd Hl and the minimum ones by even indices. In this case it is clear that is the probability of the transition S ( ) S ( ) occuring in the interval Hl H . Similarly, let Hu be an other fixed value H2k 1 H2k H2k 1 , and S ( ) the state of a hysteron at Hu . If the external field decreases now monotonically from Hu to H Hu then and naturally any one of the inequalities H2k 1 H2k 1 and Hu H dy h x,y dx 2k 1 H2k 1 can be valid. Now, let us define the random function ( )(H H y d 2k 1 H) wd Hu H 10 which gives the number of hysterons in the state S ( ) at the Hu dy h x,y dx decreasing external field H belonging to the interval y H ( ) 2k 1 H2k . Similarly, denote by u (H2k H) the num- ber of hysterons in the state S ( ) at the increasing external gives the probability of the transition S ( ) S ( ) occuring field H belonging to the interval in the interval H H 2k H2k 1 . u H . We suppose that at the starting point of the magnetizing process each hysteron is in the S ( ) state, that is the system B. Stochastic magnetizing process is in the state of negative saturation. In the following this fact Let us introduce the ``time parameter'' t 0, and will be expressed by the stochastic equation ( ) start 0. Evi- define a real valued, external field function H(t) which con- dently any other state of the system could as well be chosen sists of monotone increasing and decreasing sections of dif- for the starting point, this choice, however, does not really ferent length. Denote by H matter since the influence of the starting state on the evolu- 1 ,H2 , . . . ,H j , . . . , the extre- mum values of the function H(t) belonging to the tion of the process - as it will be shown - disappears very subsequent time points t rapidly. 1 t2 *** t j ***. It is clear that if H(t In order to describe the magnetizing process we should j) H j is a local maximum then H(t j 1) H j 1 and H(t determine two probabilities. One of them is j 1) H j 1 must be local minimums which are not nec- essarily equal. In the following the sequence Hj will be called magnetizing path and the elements of this sequence ( ) (d) ( ) are called points of reversal. If the functions H(1)(t) and P d H2k 1 H n2k 1 H start 0 H(2)(t) have the same magnetizing path then they are said to p(d) H (d) H 0 11 be equivalent for any magnetizing process irrespective of the 2k 1 2k 1 H,n2k 1 form of the time function between the individual extrema. In Fig. 2 two equivalent H(t) functions are seen. The sequence and the other is PRE 61 STOCHASTIC MODEL OF HYSTERESIS 3493 P ( ) (u) ( ) (u) (d) u H2k H n2k H start 0 exactly n2k (H) n2k 1(H2k 1) hysterons of state S ( ) p(u) (u) have to transform to the state S ( ) in the interval H2k H . 2k H2k H,n2k H 0 . 12 The probability of this event is given by It is important to note that the reversal points extremum p(u) (u) (d) values 2k H2k H,n2k H n2k 1 H2k (d) H H 2k 1 ,H2 , . . . ,H2k 1 ,H2k ,H2k 1 , . . . , N n2k 1 n(u) (d) 2k H n2k 1 H2k are Markov points of the stochastic processes ( ) u (H j H) (u) (d) and ( ) w (H) n2k 1(H2k) d (H j 1 H), and therefore we can write the following u H2k H n2k equations: (u) 1 w (H) u H2k H N n2k . 16 p(d) (d) 2k 1 H2k 1 H,n2k 1 H 0 In order to symplify the further calculations let us intro- N duce the generating functions p(d) 2k 1 (u) n (d) 2k (H2k 1) 0 2k 1 H2k 1 H,z H (d) (u) (u) N 2k 1 H,n2k 1 H n2k H2k 1 p2k H2k p(d) (d) (d) (H) 2k 1 H2k 1 H,n2k 1 H 0 zn2k 1 (d) H (u) 2k 1 ,n2k H2k 1 0 13 n2k 1(H) 0 and 17 p(u) (u) and 2k H2k H,n2k H 0 N N (u) (u) (u) (u) p(u) H p H H 0 zn2k (H). 2k 2k 2k H,z 2k 2k H,n2k (u (d) ) n n (H) 0 2k 1(H2k) 0 2k 18 H (u) (d) (d) 2k H,n2k H n2k 1 H2k p2k 1 By using the Eqs. 13 and 15 we get the first fundamental H (d) 2k 1 H2k ,n2k 1 H2k 0 . 14 equation in the form As it has already been mentioned the hysterons can be (d) (u) 2k 1 H2k 1 H,z 2k H2k H2k 1 ,a H2k 1 ,H,z , regarded as independent of each other particles and, there- 19 fore, it is an easy task to determine the probability that the number of S ( ) hysterons is exactly equal to a non-negative where integer not larger than N, at an either decreasing or increas- ing external field H provided that the number of S ( ) hys- a H2k 1 ,H,z wd H2k 1 H 1 wd H2k 1 H z. terons is known at the last reversal point before arriving at H. 20 The probability p(d) (d) (u) 2k 1 H2k 1 H,n2k 1(H) n2k (H2k 1) The second fundamental equation follows from the relations can be obtained as a result of the following consideration. If 14 and 16 . We have the number of S ( ) hysterons at the reversal point H2k 1 is equal to n(u) (d) (u) (d) 2k (H2k 1), then - in order to have n2k 1(H) 2k H2k H,z c H2k ,H,z N 2k 1 hysterons in the state S ( ) at the external field H H2k 1 - exactly n(u) (d) H2k 1 H2k ,b H2k ,H,z , 21 2k (H2k 1) n2k 1(H) hysterons of state S ( ) have to transform to the state S ( ) in the interval where H2k 1 H . It is obvious that the probability of this event can be given by c H2k ,H,z 1 1 z wu H2k H 22 p(d) (d) (u) 2k 1 H2k 1 H,n2k 1 H n2k H2k 1 and (u) H2k 1 z n2k (u) (d) w (H2k 1) n2k 1(H) n(d) d H2k 1 H n2k b H2k ,H,z . 23 2k 1 H c H2k ,H,z (d) 1 w (H) d H2k 1 H n2k 1 . 15 Now we will derive the characteristic function of the probability that the system magnetization is not larger than x Similarly, to determine the probability at a decreasing external field H which follows the last rever- p(u) (u) (d) 2k H2k H,n2k (H) n2k 1(H2k) one has to recognize that sal point H2k 1. Introducing the notation if the number of S ( ) hysterons at the reversal point H2k is equal to N n(d) (u) 2k 1(H2k), then - in order to have n2k (H) hysterons in the state S ( ) at the external field H H 24 2k - 3494 L. PAŽL PRE 61 and by using the relation 17 we obtain from Eq. 4 where (d) (u) d H2k 1 H, N 2k 1 H2k 1 H, . H (u) 2k H,z 25 N2k H2k H 0 d 2k dz . 31 z 1 Similarly, if we use Eq. 18 then the characteristic function of the probability that the magnetization of the system is not The recursive relation larger than x at an increasing external field H after the last N(u) (d) reversal point H 2k H2k H 0 Nwu H2k H N2k 1 H2k 1 H2k 0 2k can be obtained from Eq. 4 in the form 1 w (u) u H2k H 32 u H2k H, N 2k H2k H, . 26 follows from the other fundamental Eq. 21 . These two characteristic functions describe completely the By intoducing the relative magnetizations stochastic behavior of the magnetizing process in both in- creasing and decreasing external magnetic fields. It is appar- 1 ent from the above considerations that the stochastic model m(d) (d) 2k 1 H2k 1 H 0 NM M2k 1 H2k 1 H 0 33 developed by us has been built up without any reference to a s particular nature of hysteresis and therefore, its generality is and at least as high as that of the Krasnoselskii and Pokrovskii 8 model. 1 m(u) (u) 2k H2k H 0 NM M2k H2k H 0 , 34 s C. Calculation of the hysteresis curves from Eqs. 27 , 29 and 30 , 32 after elementary calcula- The expectation value of the magnetic moment k due to tions the following recursive relations are obtained: the kth hysteron can be given by m(d) 2k 1 H2k 1 H 0 E k i 1 d d Ms . 1 m(u) H 0 2k 2k H2k 1 0 1 wd H2k 1 H 1 By using this expression we can write the expectation value 35 of the magnetization of the system at the decreasing external and field H following the reversal point H2k 1 in the form m(u) 2k H2k H 0 i 1 d d H2k 1 H, (d) d 2wu H2k H 1 m2k 1 H2k 1 H2k 0 0 M(d) 1 wu H2k H 1. 36 2k 1 H2k 1 H 0 2M (d) In order to solve this system of recursive equations we sN2k 1 H2k 1 H 0 NM s , 27 need the formula for the starting branch of the relative mag- where netization. Since the negative saturation has been chosen as the initial state of the system it follows from Eq. 9 that if (d) H Hl H0 , then N(d) 2k 1 H,z 2k 1 H2k 1 H 0 d 2k 1 dz . z 1 w 28 u Hl H Fu H C , and so the equation for the starting branch will be From the fundamental Eq. 19 we obtain m(u) H 0 2F N(d) 0 u H C 1. 37 2k 1 H2k 1 H 0 N(u) This branch can be also called the limiting ascending branch 2k H2k H2k 1 0 1 wd H2k 1 H . because there is no branch below it. If the positive saturation 29 would be the initial state then it is easy to show that the limiting descending branch can be written in the form The expectation value of the magnetization of the system at the external magnetic field H increasing after the reversal m(d) 0 H 0 2Fd H C 1, 38 field H2k can be obtained from the equation where Fd(H C) is defined by the expression 8 , and at the same time it is obvious that the limiting descending branch i 1 d u H2k H, d has the property that there is no other branch above it. The 0 two limiting curves form the major hysteresis loop which M(u) 2k H2k H 0 defines an area where all other loops should be located. By using the expression 37 for the starting branch and 2M (u) sN2k H2k H 0 NM s , 30 Eq. 35 we can obtain the first descending branch PRE 61 STOCHASTIC MODEL OF HYSTERESIS 3495 m(d) (u) 1 H1 H 0 2Fu H1 C 1 wd H1 H 1, 39 m2 Hd H 0 2Fu Hu C 1 wd Hu Hd which is attached to the limiting ascending branch at the 1 wu Hd H 2wu Hd H 1 point H1. This descending branch is called by Mayergoyz 1 44 the first-order transition curve. The field H1 where the first- order transition curve starts from, will be called start field. for the ascending branch of the same loop. It is to note that Denote by H2 the next reversal point where the magnetizing according to the Mayergoyz's terminology the first minor field begins again to increase. The corresponding ascending loop consists of a first-order descending and a second-order branch, i.e., the second-order transition curve is given by the ascending transition curves. Following the reversal points of formula the magnetizing field this procedure can be continued and we can obtain both the descending and ascending branches of m(u) 2 H2 H 0 2wu H2 H relative magnetization for any minor loop. One can prove a very important limit theorem, namely, 1 m(d) 1 H1 H2 1 wu H2 H 1. there exist two limit curves 40 lim m(d) 2k 1 Hu H 0 md Hu H , 45 This procedure can be continued and it is seen that there is k no need to take into account any special requirement in order to describe the field dependence of the average magnetiza- and tion since the Markov points of the magnetizing field deter- mine exaclty the stochastic behavior of the process. lim m(u) 2k Hd H 0 mu Hu H , 46 k D. Stationarity of hysteresis loops which are determining a closed minor loop. In other words, Let us investigate now the variation of the magnetization the magnetizing process becomes stationary with increasing for a special sequences of reversal fields. Let as suppose that number of cycles. It means the system ``forgets'' gradually H2k 1 Hu , k 0,1, . . . , while H2k Hd , k 1,2, . . . , its initial state by repeating the magnetizing cycle. This for- and Hu Hd , i.e., the magnetizing field is varying between getting process can be related to the well-known accommo- two extreme values Hu and Hd . The field variation which dation process. The original Preisach model results in an starts with a decrease of the the external magnetic field H immediate formation of the minor hysteresis loop after only from the reversal point Hu until it reaches the next reversal one cycle of back-and-forth variation of the input between point Hd and then turns to increase to the nearest Hu value, any two consecutive extremum values. However, this conse- is called the magnetizing cycle. The magnetizing cycle re- quence of the Preisach model contradicts to well known ex- sults in a hysteresis loop called the minor hysteresis loop. perimental finding that the hysteresis loop formation is pre- The first cycle corresponds to the variation of the external ceded by an accommodation process which can be field between the reversal points H1 H2 H3, where H1 sometimes appreciable 10,11 . In order to describe this ac- H3 Hu and H2 Hd , while kth cycle is done by the commodation process the traditional Preisach model was variation of the magnetizing field between the reversal points modified in the ``moving'' and the ``product'' models 12 . H One has to mention that the modification of the moving 2k 1 H2k H2k 1, where H2k 1 H2k 1 Hu and H2k H model performed by Mayergoyz 1 see Sec. II. 5, pp. 108­ d for k 1,2, . . . . For the descending branch of the kth minor loop one can obtain from Eq. 35 the following ex- 114 gives not only a possible variant of the accommodation pression: but defines a sufficient condition too for the convergence of the process. The stochastic model developed by us contains m(d) 2k 1 Hu H 0 the phenomenon of accommodation inherently as a conse- quence of the limit theorem 45 and 46 . 1 m(u) 2k 2 Hd Hu 0 1 wd Hu H 1. The formulas for the limit curves defined by Eqs. 45 and 41 46 can be obtained by some elementary calculations as fol- lows: while for the ascending branch of the kth minor loop the relation md Hu H 2Q Hd ,Hu wu Hd Hu 1 wd Hu H 1 47 m(u) 2k Hd H 0 2wu Hd H and 1 m(d) 2k 1 Hu Hd 0 m 1 w u Hd H 1 2Q Hd ,Hu wd Hu Hd 1 wu Hd H , u Hd H 1 42 48 can be derived from Eq. 36 . By using Eq. 37 we have where m(d) 1 Hu H 0 2Fu Hu C 1 wd Hu H 1, 43 Q Hd ,Hu wd Hu Hd wu Hd Hu for the descending branch of the first minor loop and wd Hu Hd wu Hd Hu 1. 3496 L. PAŽL PRE 61 From these equations two important relations can be derived, namely, md Hu Hu mu Hd Hu and md Hu Hd mu Hd Hd , which show that the return-point memory property is fulfiled for the accommodated minor hysteresis loops. It is also ob- vious, that the accommodated minor loops due to the same pair of reversal fields Hd and Hu Hd are not only congruent but identical since the field values Hd and Hu unambigu- ously determine the the branches of stationary loops. One has to mention that the accommodated branches md( H) and mu( H) are exactly identical with the limiting de- scending and ascending branches which indicates the consis- tency of the theory. FIG. 3. Contour plot of h(x,y C) defined by Eq. 6 where The explicit form of the expressions md(Hu H) and h(x,y) is given by Eq. 54 with parameter values Hc 0.2, mu(Hd H) which describe the descending and the ascending 0.6, and Cr 0.5. branches of the stationary minor loop between two reversal fields Hd and Hu Hd has a great advantage in numerical calculations in comparision with the well-known Everett in- h x,0 dx tegral. It is to be noted that the expressions 47 and 48 are dm 0 0 H 53 suitable to describe not only symmetrical but asymmetrical a lim x H 0 dH hysteresis loops too and it is easy to show that symmetrical dx h x,y dy 0 hysteresis loops can be obtained only if the function h(x,y) has a mirror symmetry expressed by is different from zero in contrary to the classical Preisach model which gives a nonrealistic zero slope of the virgin h x,y h y, x . 49 curve at H 0. In the following the mirror symmetry of h(x,y) will be as- IV. NUMERICAL CALCULATIONS AND DISCUSSION sumed. It is worthwhile to derive the formula for the virgin curve In order to compute the magnetization vs field curves we of the magnetization depending on the parameters of the den- have to know the joint density function h(x,y C) of the U sity function h(x,y C). After some simple manipulations we and D fields. Since these fields are the sum of many small obtain random components it is reasonable to assume that the cen- tral limit theorem is approximately valid and so the function s h(x,y) in h(x,y C) can be chosen in the form m 1 H 0 H 2s 1, 50 1 H s2 H s1 H s2 H 1 1 h x,y exp x H 2 2 c 2 where 2 2 1 Cr 2 2 1 Cr H x dx h x,y dy 2Cr x Hc y Hc y Hc 2 , 54 H s1 H 51 x dx h x,y dy where the meaning of the constants Hc , , and Cr is clear H from the elements of the probability theory. Figure 3 shows the contour plot of h(x,y C) defined by Eq. 6 for the pa- and rameters Hc 0.2, 0.6, and Cr 0.5. The contours are be- longing to the following values of h(x,y C) 0.1, 0.2; H 0.3 0.05 0.65 and 0.682. The last one is slightly smaller than dy h x,y dx max H y (x,y)h(x,y C) 0.682923***. The discontinouity along the s line y x 0 can be clearly seen in the figure. In the sequel 2 H . 52 H dy h x,y dx this formula will be used in all of our numerical calculations y provided that the correlation coefficient Cr is equal to zero, i.e., h(x,y) f (x) f ( y) where f is the density function of If the function h(x,y) satisfies the symmetry relation 49 the normal distribution. This case corresponds to the product then it is easy to prove that model introduced by Biorci and Pescetti 13 and used con- sequently by KaŽdaŽr 5,6,12 . lim m The relative magnetization vs field curves are shown in 0 H 0 H 0 Fig. 4. The parameter values used for the calculation are Hc 0.4, 0.6, and Cr 0. The values of magnetizing field and the initial susceptibility defined by are given here and in the further figures in a properly chosen PRE 61 STOCHASTIC MODEL OF HYSTERESIS 3497 FIG. 4. Limiting branches LA, LD, and magnetization versus FIG. 6. Convergence of the relative magnetization in the rever- field curves starting from the reversal points H1 1.2, H2 0.8, sal point Hu 0.8 with increasing number of cycles to the limit i.e., H3 0.7, and H4 0.6. The magnetization curves are indicated by the stationary value. 1, 2, 3, and 4. from, occupies a higher position than the point A, and the arbitrary unit. The curves LA and LD correspond to the lim- end point C of the increasing branch of the second minor iting ascending and descending branches, while the curves loop is found above the point B but the distance between the indexed in the figure by 1, 2, 3, 4 are the first-, second-, points C and B is smaller than that between the points B and third-, and fourth-order transition curves defined by the re- A. By repeating the magnetizing cycle between the reversal versal points H fields Hd 0.2 and Hu 0.8 the difference between the 1 1.2,H2 0.8,H3 0.6,H4 0.6. It is worth noting that the all information about the past history of branches of the same type becomes gradually negligible, i.e., the branches converge to limit curves which form finally a the magnetizing process is transfered by the state of the sys- tem in the last reversal point. For example, the fourth-order closed stationary hysteresis loop denoted by LC. The mag- transition curve 4 which is plotted in the field interval netizing curves measured by Carter and Richards 11 on 0.6,1.5 , is determined by the state in the reversal point silicon steel 4.3% Si are surprisingly similar to that plotted H in Fig. 5. 4 0.6. It is well known that in the traditional Preisach model the In order to demonstrate the speed of the convergence, the minor loops which describe the cyclic change of the magne- nonaccommodated relative magnetizations have been calcu- tization with back-and-forth variation of the magnetizing lated in the reversal point Hu 0.8 for the subsequent cycles. field between the same two limiting values are congruent and Figure 6 shows that the stationary i.e., the limit value of the the formation of a closed minor loop is realized in one cycle, magnetization can be very well approached by repeating the i.e., the accommodation process is absent. In contrast to this cycle 8­9 times in the case of parameter values Hc 0.4, the stochastic model contains inherently the accommodation 0.6, and Hu 0.3. process which is clearly demonstrated in Fig. 5. For the sake The nonaccommodated minor loops due to the same pair of orientation the limiting ascending branch LA is also plot- of reversal fields are evidently not congruent and generally ted in Fig. 5 where it is seen that the descending branch of are not closed. However, this noncongruency has nothing in the first minor loop starts from the point A due to the first common with that introduced and discussed in details by reversal field H KaŽdaŽr 5,15 . The noncongruency of the nonaccommodated 1 0.8 and after reaching the reversal point H minor loops bounded by the same field limits has a quite 2 Hd 0.2 it turns to increase to the point B which corresponds to the next reversal field H different origin in the stochastic model, namely, the nonequi- 3 Hu 0.8. One can observe that the first minor loop is not closed, the point B librium response of the system for the cyclic back-and-forth where the decreasing branch of the second minor loop starts variation of the external magnetic field between two con- secutive reversal points. It is obvious consequence of the non-stationarity of minor loops that the return-point memory property is absent in these loops. In order to study the properties of noncongruency of this type the first minor loops belonging to different start fields are calculated. Denote by H Hu Hd the difference be- tween the consecutive reversal fields. For the characteriza- tion of the nonaccommodated first minor loops due to differ- ent start fields H1 let us introduce two parameters defined by W W H1 , H max m1 H1 H H1 H H H1 m2 H1 H H ] FIG. 5. Accommodation process of the minor loop in consecu- tive magnetizing cycles between the reversal points Hd 0.2 and Hu 0.8. The loop LC is the accommodated minor loop. and 3498 L. PAŽL PRE 61 FIG. 7. Width W of the first-order minor loops and the differ- FIG. 9. The nonaccommodated NR and the stationary SR rela- ence O between the values of the descending and ascending tive remanences versus start field due to different points of the branches in the reversal point H ascending limiting branch. u 0.8 versus start field. O O H It seems to be useful to calculate the accommodated (sta- 1 , H m2 H1 H H1 m1 H1 H1 . tionary) hysteresis loops for different pairs of reversal points The dependence of these parameters on the start field H H 1 is d and Hu Hd . The hysteresis loops plotted in Fig. 11 shown in Fig. 7 for the parameter values H correspond to the reversal points H d 0.2, H d 1.5,Hu 1.5 loop 1. The author of the present paper is far not convinced ML1 , Hd 1,Hu 1 loop ML2 , Hd 0.5,Hu 0.5 whether the experimental data contradict or support the non- loop ML3 . For the calculation we used the parameter val- congruency of this type because of the lack of careful mea- ues Hc 0.4 and 0.6. For the sake of completeness the surements. virgin curve VC calculated by Eq. 50 and the major loop It seems to be useful to investigate the remanence prop- LL bounded by the limiting ascending and descending erties of systems described by the stochastic model. In Fig. 8 curves are also shown in the figure. The stochastic model the first-order descending curves which start from different clearly shows that all accommodated minor loops corre- points of the ascending limiting branch LA can be seen. The sponding to cyclic inputs between the same two consecutive curves starting from the points due to the field values H extremum values are not only congruent but simply identical. 1 1.6,H In Fig. 12 three accommodated first-order minor loops 2 1.4,H3 1.2,H4 1 are plotted to the points of remanences R1, R2, R3, R4 which are obviously different denoted by 1, 2, 3 can be seen. The descending branches of from the stationary i.e., the accommodated values. The the loops are started from the field values H 0.5, 0.3, 0, and nonaccommodated NR and stationary remanences SR versus each of the ascending branches returns exactly to the same start field are shown in Fig. 9. As it is seen the nonaccom- point that the corresponding descending branch left. The re- modated remanences can be negative below a critical start turning curves have an apparent slope discontinuity with re- field CR since the initial negative saturation has a significant gard to the major loop ALA. effect on the first-order transition curves. The stationary re- At this point it is worth to make a remark of somewhat manence curve SR calculated from Eqs. 47 and 48 is historical nature. As it is well-known Preisach's idea for his non-negative in all points of the start field interval. model was originated from the quadratic Rayleigh relation The influence of the parameter on the shape of the which can be easily obtained 14 assuming a uniform dis- major hysteresis loop can be seen in Fig. 10. As it is ex- tribution of the U and D fields over the ``Preisach triangle.'' pected the larger is the parameter the wider is the hyster- It is interesting to note that in the stochastic model the cal- esis loop, i.e., the larger nonhomogeneity in a system e.g., in culated hysteresis loops almost perfectly coincide with that a magnetic sample results in a higher ``coercive force.'' calculated by the Rayleigh formula when the reversal fields Hd and Hu Hd and so the magnetizing field H Hd ,Hu FIG. 8. The limiting ascending branch LA and four first-order descending curves ending in nonaccommodated remanences de- FIG. 10. Influence of the parameter on the shape of the major noted by R1, R2, R3, and R4. hysteresis loop in the case of Hc 0.4. PRE 61 STOCHASTIC MODEL OF HYSTERESIS 3499 FIG. 11. The virgin curve VC, the major loop LL and three FIG. 13. The hysteresis loop between ''small'' reversal points accommodated hysteresis loops ML1, ML2, ML3 calculated for and the quadratic Rayleigh curves denoted by squares. different pairs of reversal points in the case of parameter values a Dirac-delta-like singularity along the boundery of the Prei- Hc 0.4 and 0.6. sach triangle, as it was shown by Mayergoyz 1 . The sus- are sufficiently small. The hysteresis loop R defined by re- ceptibility vs magnetizing field is seen in Fig. 14 for the versal points H parameter values H u 0.5 and Hd 0.5 in Fig. 13 can be very c 0.2 and 0.6. The shape of the cal- well approximated by the equations culated curve can be expected on the basis of physical con- siderations and corresponds to those found experimentally. md (d) (d) (d) a 0.5 H C0 C1 H C2 H2, The estimation of the joint density function h(x,y C) from measured hysteresis curves was beyond the scope of mu (u) (u) (u) a 0.5 H C0 C1 H C2 H2, our present theoretical consideration. Of course, one may attempt in simple cases to estimate the parameters of a plau- where sible density function e.g., Eq. 54 by an appropriate data evaluation procedure. C(d) (u) 0 C0 0.13035***, V. CONCLUSIONS C(d) (u) 1 C1 0.79301***, It has been shown that the Preisach model can be im- C(d) (u) proved by describing the hysteresis as a stochastic process 2 C2 0.54251*** defined on a set of all possible values of the control param- in the case of parameter values H eter the reversal turning points of which are Markov points c 0.2 and 0.6. In Fig. 13 the squares correspond to the values calculated by the of the process. The one dimensional distribution function of quadratic equations. The excellent aggreement with the the stochastic process has been exactly determined and the curves of the stochastic model indicates that the Rayleigh magnetizations versus up and down magnetic fields have law can be reproduced in a straightforward way in the sto- been calculated as expectation values of the stochastic pro- chastic model. cess. It has been proven that the magnetizing process be- This model differs from the original Preisach model in a comes stationary with increasing number of magnetizing very essential point in relation to the reversal point suscep- cycles. It means that for the description of the accommoda- tibility. Namely, the nonzero initial susceptibility at the turn- tion process there is no need of any artificial auxiliary as- ing points is an inherent property of the stochastic model, sumption since the stochastic model contains the phenom- while the traditional Preisach model can produce positive enon of accommodation inherently. In general case the initial slope only if the Preisach function is supposed to have model is able to describe the symmetric as well as the asym- metric hysteresis. In relatively small magnetizing fields the FIG. 12. Three accommodated first-order minor loops denoted FIG. 14. The irreversible susceptibility versus magnetizing by 1, 2, 3 and the shifted ascending branch ALA. field H. 3500 L. PAŽL PRE 61 quadratic Rayleigh law can be easily obtained from the equa- but the nonstationary loops are noncongruent and in general tions of the stochastic model. It is important to note that the not closed. turning point susceptibilities have nonzero finite values in contrary to the traditional Preisach model which does not ACKNOWLEDGMENT take consequently into account the random nature of the el- ementary switching process. Finally, the stochastic model The author wishes to acknowledge stimulating discus- shows that all stationary loops corresponding to the same sions with Mr. G. KaŽdaŽr, who helped greatly in completing two limiting values of the magnetizing field are equivalent this paper. 1 I. D. Mayergoyz, Mathematical Models of Hysteresis 11 R.O. Carter and D.L. Richards, J. Am. Ceram. Soc. 97, 199 Springer-Verlag, Berlin, 1991 . 1950 . 2 A. Visintin, Differential Models of Hysteresis Springer- 12 E. Della Torre and G. KaŽdaŽr, IEEE Trans. Magn. 23, 2823 Verlag, Berlin, 1994 . 1987 . 3 A. IvaŽnyi, Hysteresis Models in Electromagnetic Computation 13 G. Biorci and D. Pescetti, Nuovo Cimento 7, 829 1958 . AkadeŽmiai KiadoŽ, Budapest, 1997 . 14 R. Becker and W. Došring, Ferromagnetismus Springer, Ber- 4 G. Bertotti, Hysteresis in Magnetism Academic, San Diego, lin, 1939 in German , p. 222. 1998 . 15 G. KaŽdaŽr and E. Della Torre, IEEE Trans. Magn. 23, 2820 5 G. KaŽdaŽr, J. Appl. Phys. 61, 4013 1987 . 1987 . 6 G. KaŽdaŽr, Phys. Scr. T25, 161 1989 . 16 After the first submission of this manuscript a very interesting 7 F. Preisach, Z. Phys. 94, 277 1935 . paper was published: G. Bertotti, I.D. Mayergoyz, V. Basso, 8 M.A. Krasnoselskii and A.V. Pokrovskii, Sov. Math. Dokl. 12, and A. Magni, Phys. Rev. E 60, 1428 1999 about the func- 1388 1971 . 9 L. PaŽl, Foundation of the Probability Calculus and Statistics tional integration approach to hysteresis over an abstract prob- AkadeŽmiai KiadoŽ, Budapest, 1995 in Hungarian , Vol. 1, p. ability space of Kolmogorov. This approach is different from 122. that which is described in present paper and has certainly a 10 W.S. Melville, J. Inst. Electron Eng. 97, 165 1950 . much wider field of possible applications.