Nonequilibrium Statistical Mechanics of Noninteracting Classical

Journal of the Korean Physical Society, Vol. 52, No. 6, June 2008, pp. 17151725

Nonequilibrium Statistical Mechanics of Noninteracting Classical Spins in a Rotating Magnetic Field Suhk Kun Oh BK21 Physics Program, Department of Physics and Basic Science Research Institute, Chungbuk National University, Cheongju 361-763

(Received 12 November 2007, in nal form 14 March 2008) A model of noninteracting classical spins in a time-dependent rotating magnetic eld is investigated. The nonequilibrium magnetization, equal-time correlation functions, time-displaced correlation functions, and the energy are calculated to see their dependences on the magnitude of a transverse rotating magnetic eld and a rotating angular frequency. For small rotating elds, compared to a static eld along the z -axis, all the nonequilibrium quantities show remarkable increases in the oscillation amplitudes at the resonance frequency. However, due to the competition between the static eld and the rotating eld, the oscillation amplitude increases monotonically for large rotating elds as we increase the angular frequency beyond the resonance angular frequency and no resonance phenomena are seen.

PACS numbers: 02.70.Uu, 05.10.Ln, 05.50.+q, 05.70.Jk, 05.740.Ln, Keywords: Nonequilibrium spin dynamics, Rotating coordinate system, Time-dependent correlation functions I. INTRODUCTION

Unlike equilibrium statistical mechanics where the Gibbs ensemble approach [1] has been tremendously successful, there is no generally accepted method to investigate nonequilibrium statistical mechanics problems [2{ 4]. The maximum entropy formalism developed by E. T. Jaynes and others [5,6] seems to be a feasible candidate, but the calculation of the partition functional with given constraints is a formidable task for us to proceed further except for some extremely simple cases. Thereby, even though dynamical behavior away from equilibrium is vastly dierent from that in equilibrium, many people have a tendency to adopt the Green-Kubo linear response formalism [7,8] to study nonequilibrium problems without proper assessment. One of the simplest nonequilibrium problems is the nonequilibrium behavior of a system with a given Hamiltonian. Here, we usually adopt the equilibrium states as our initial conditions and let each equilibrium con guration evolve in time as dictated by the equations of motion. Nevertheless, even if we are equipped with an exact formalism to study this kind of problem, we still face the notoriously dicult problem of solving equations of motion. Since exactly solved problems are rare, it would even be meaningful to obtain the full solution to a very simple model and then to closely examine the consequencies to understand the nonequilibrium behavior of

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a system. For this purpose, we investigate a simple model of noninteracting classical spins in a time-dependent rotating eld, which originated from nuclear magnetic resonance studies of magnetic materials [9]. For this model, the equations of motion for the nonequilibrium ensemble averages of the total spin components are the so-called phenomenological undamped Bloch equations [10{13]. Even for these undamped Bloch equations, only approximate solutions in the small transverse eld limit were given [8]. Hence, in this work, we would like to provide not only the exact solutions of the undamped Bloch equations of motion with arbitrary transverse eld strength but also the interesting dependence of other nonequilibrium quantities like equal-time correlation functions, time-displaced correlation functions and the time-dependent energy, on the magnitude of transverse rotating magnetic elds and rotating angular frequencies. II. THE MODEL

Our model of N noninteracting classical spins in a rotating magnetic eld is de ned by the following Hamiltonian:

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H= ! h (t)

N X !s i=1

i

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= hz

N X sz i=1

i

h0

N X (sx cos !t i=1

i

Journal of the Korean Physical Society, Vol. 52, No. 6, June 2008

syi sin !t);

(1)

where ! h (t) denotes the time-dependent magnetic eld, hz is its component along the z -direction and h0 is the magnitude of its components rotating clockwise with angular frequency !. Furthermore, sxi = sin i cos i , y si = sin i sin i and szi = cos i are the components of classical spin !s i at site i. Here, i and i denote the polar and the azimuthal angles, respectively. In terms of P N the total spin components, S = i=1 si ( = x; y; z), the above Hamiltonian can be expressed in the form H= ! h (t) ! S = hz Sz h0 (Sx cos !t Sy sin !t): (2) Here, the usual!gyromagnetic ratio is absorbed in the magnetic eld h (t). III. EQUATIONS OF MOTION AND THEIR SOLUTIONS

Let us study the equations of motion for the total spin vector ! S , which can be found from d! S ! = h (t) ! S: (3) dt Now, we explicitly write down the equations of motion for the total spin components as follows: dSx = hz Sy + h0 Sz sin !t; dt dSy = hz Sx + h0 Sz cos !t; dt dSz = h0 Sy cos !t h0 Sx sin !t: dt

(4) (5)

(6) As mentioned previously, nonequilibrium ensemble averaging of the above equations of motion gives rise to the phenomenological undamped Bloch equations [8,10]. The generalized version of the Bloch equations, including phenomenological relaxation terms, has been successfully used to explain both the phenomena of nuclear magnetic resonance and magnetization reversal achieved through an application of an electromagnetic pulse [9,11]. The above equations are not easy to solve directly. In order to solve them, let us consider a coordinate system rotating clockwise with respect to the z-axis with angular frequency !. Then, the time derivative of the total spin vector in the xed coordinate system and that in the rotating coordinate system are related by ! d! S jfixed = @ S jrotating + ! ! ! S= ! h (t) ! S :(7) dt

@t

In the rotating coordinate system, the explicitly timedependent terms can be removed from the equations

of motion. From now on, for the sake of notational convenience, we shall write d! S =dtjfixed as d! S =dt and ! ! @ S =@tjrotating as @ S =@t and we call our new coordinate system, the -ed coordinate system. Now, the equation of motion for the total spin vector in the -ed coordinate system can be expressed in the form @! S = [! h (t) + ! !]! S ; (8) @t

and the equations of motion for total spin components are @Sx = (hz !)Sy ; @t @Sy = (hz !)Sx + h0 Sz ; @t @Sz = h0 Sy : @t

(9) (10) (11)

Since the equations of motion are still not easy to solve, let us introduce another new coordinate system rotated by an angle with respect to the -ed coordinate system, where the p eective magnetic eld strength is given by heff = h20 + (hz !)2 and is along the new x-axis, tan = (hz !)=h0 , sin = (hz !)=heff and cos = h0 =heff . We call this coordinate system the -ed coordinate system. The equations of motion of the total spin components S ( = x; y; z) in -ed coordinate system are of the form @Sx = 0; @t @Sy = heff Sz ; @t @Sz = heff Sy ; @t

(12) (13) (14)

which are easy to solve and the solutions are of the form Sx (t) = Sx (0); (15) Sy (t) = Sy (0) cos heff t + Sz (0) sin heff t; (16) Sz (t) = Sy (0) sin heff t + Sz (0) cos heff t: (17) After performing the long and arduous algebra of rst converting these solutions and unit vectors in the -ed coordinate system into those in the -ed coordinate system and then converting the solutions and unit vectors in the -ed coordinate system into those in the original xed coordinate system, we obtain Sx (t) = Sx (0)[fcos2 + sin2 cos(heff t)g cos(!t) sin sin(heff t) sin(!t)] +Sy (0)[sin sin(heff t) cos(!t) + cos(heff t) sin(!t)] +Sz (0)[sin cos f1 cos(heff t)g cos(!t) + cos sin(heff t) sin(!t)]; (18) 2 2 Sy (t) = Sx (0)[ fcos + sin cos(heff t)g sin(!t)

Nonequilibrium Statistical Mechanics of Noninteracting { Suhk Kun Oh

sin sin(heff t) cos(!t)] +Sy (0)[ sin sin(heff t) sin(!t) + cos(heff t) cos(!t)] +Sz (0)[ sin cos f1 cos(heff t)g sin(!t) + cos sin(heff t) cos(!t)]; (19) Sz (t) = Sx (0) sin cos f1 cos(heff t)g +Sy (0) cos sin(heff t) +Sz (0)fsin2 + cos2 cos(heff t)g: (20)

Since our initial conditions (ICs) for the magnetic resonance experiments are usually prepared in equilibrium, we adopt the equilibrium states as the initial conditions for our equations of motion in our study of the nonequilibrium behavior. In order to nd the equilibrium values of the magnetization and the correlation functions from an ensemble of equilibrium states, we consider the timeindependent Hamiltonian H0 given by H0 = hz Sz h0 Sx h1 Sy X X X = hz szi h0 sxi h1 syi ; (21) i

i

where we introduced some ctitious magnetic eld h1 along the y-axis for calculational purpose. Then, the partition function for this Hamiltonian is given by

Z Z Z Z

2 Z = Ni=1 sin i di di e H0 0 0 2 =[ ea cos i +b sin i cos i +c sin i sin i 0 0 sin i di di ]Np N ( a2 + b2 + c2 ) = (4)N sinh (a2 + b2 + c2 )N=2 ;

0

0

(22)

f (a cos + b sin cos + c sin sin ) sin dd

= 2 = 2

Z p f ( a2 + b2 + c2 cos ) sin d 0 Z1 p 1

f (( a2 + b2 + c2 z )dz;

< Sx >eq N b = p 2 2 coth a2 + b2 a +b < Sy >eq meq = 0; y = N < Sz >eq meq z = N a = p 2 2 coth a2 + b2 a +b meq x =

p

(23)

where is some angle with respect to the z-axis in a new coordinate system rotated in three dimensions from the original coordinate system. After carrying out appropriate partial dierentiations and setting c = 0, we get the magnetization and the equilibrium correlation functions for the total spin components in the absence of h1 eld as follows:

b ; a2 + b2

(24) (25)

a : a2 + b2

(26)

(2) Equilibrium Correlation Functions The equilibrium correlation functions, which are necessary for us to calculate the time-dependent correlation functions in nonequilibrium, are given by < Sx >2eq

< Sx2 >eq

1 b2 p = f g 2 N a2 + b2 (a + b2 )3=2 2 p p coth a2 + b2 a2 b+ b2 csch2 a2 + b2

a2 b2 (27) (a2 + b2 )2 ; < Sy2 >eq = p 21 2 coth a2 + b2 a2 +1 b2 ; (28) N a +b 2 < Sz >eq < Sz >2eq 1 a2 p = f g N a2 + b2 (a2 + b2 )3=2 2 coth a2 + b2 a2 a+ b2 csch2 a2 + b2 2 2 + (aa2 + bb2 )2 ; (29) < Sx Sz >eq < Sx >eq < Sz >eq N 2 ab = (a2 + b2 )2 (a2 +abb2 )3=2 coth a2 + b2 ab csch2 a2 + b2 ; (30) 2 a + b2 < Sx Sy >eq = 0; (31) < Sy Sz >eq = 0: (32)

p

p

p

where a = hz , b = h0 , c = h1 and is the inverse temperature and where we have made use of the formula [14]

Z 2Z

(1) Equilibrium Magnetization The magnetization components meq < S >eq =N ( = x; y; z ) in equilibrium are given by

p

IV. EQUILIBRIUM PROPERTIES

i

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p

p

(3) Equilibrium Energy per Spin Eeq = hz meqz heq0 meqx N = hz [ p 2a 2 coth a2 + b2 a2 +a b2 ] a +b b b h0 [ p 2 2 coth a2 + b2 ]: a2 + b2 a +b

p

p

(33)

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V. EXACT EXPRESSIONS FOR THE MAGNETIZATION, THE TIME-DEPENDENT CORRELATION FUNCTIONS AND THE ENERGY IN NONEQUILIBRIUM

Making use of the solutions for the total spin components of the previous section, we calculate the exact expressions for the magnetization, thew time-dependent correlation functions and the energy in nonequilibrium. (1) Magnetization In term of the equilibrium magnetization components given in Eq. (18), the magnetization components are given by 2 2 mx (t) = meq x [fcos + sin cos(heff t)g cos(!t) sin sin(heff t) sin(!t)] +meqz [sin cos f1 cos(heff t)g cos(!t) + cos sin(heff t) sin(!t)]; (34) 2 eq 2 my (t) = mx [ fcos + sin cos(heff t)g sin(!t) sin sin(heff t) cos(!t)] +meqz [ sin cos f1 cos(heff t)g sin(!t) + cos sin(heff t) cos(!t)]; (35) eq mz (t) = mx sin cos f1 cos(heff t)g +meqz fsin2 + cos2 cos(heff t)g: (36) (2) Equal-time Correlation Functions If the equilibrium correlation functions given in Eq. (19) are used, the equal-time correlation functions are given by < [Sx (t)]2 >=< Sx2 >eq [fcos2 + sin2 cos(heff t)g cos(!t) sin sin(heff t) sin(!t)]2 + < Sy2 >eq [sin sin(heff t) cos(!t) + cos(heff t) sin(!t)]2 + < Sz2 >eq [sin cos f1 cos(heff t)g cos(!t) + cos sin(heff t) sin(!t)]2 + < Sx Sz >eq [fcos2 + sin2 cos(heff t)g cos(!t) sin sin(heff t) sin(!t)] [sin cos f1 cos(heff t)g cos(!t) + cos sin heff t sin(!t)]; (37) 2 2 < [Sy (t)] >=< Sx >eq [ fcos2 + sin2 cos(heff t)g sin(!t) sin sin(heff t) cos(!t)]2 + < Sy2 >eq [ sin sin(heff t) sin(!t) + cos(heff t) cos(!t)]2 + < Sz2 >eq [ sin cos f1 cos(heff t)g sin(!t) + cos sin(heff t) cos(!t)]2 + < Sx Sz >eq [ fcos2 + sin2 cos(heff t)g sin(!t) sin sin(heff t) cos(!t)]

[ sin cos f1 cos(heff t)g sin(!t) + cos sin(heff t) cos(!t)]; (38) 2 2 2 2 < [Sz (t)] >=< Sx >eq sin cos f1 cos(heff t)g2 + < Sy2 >eq cos2 sin2 (heff t) + < Sz2 >eq [sin2 + cos2 cos(heff t)]2 + < Sx Sz >eq sin cos [1 cos(heff t)] [sin2 + cos2 cos(heff t)]; (39) (3)Time-displaced Correlation Functions If the equilibrium correlation functions given in Eq. (19) are used, the time-displaced correlation functions are given by < Sx (t)Sx (0) >=< Sx2 >eq [fcos2 + sin2 cos(heff t)g cos(!t) sin sin(heff t) sin(!t)] + < Sx Sz >eq [sin cos f1 cos(heff t)g cos(!t) + cos sin(heff t) sin(!t)]; (40) 2 < Sy (t)Sy (0) >=< Sy >eq [ sin sin(heff t) sin(!t) + cos(heff t) cos(!t)]; (41) 2 2 2 < Sz (t)Sz (0) >=< Sz >eq [sin + cos cos(heff t)] + < Sx Sz >eq sin cos [1 cos(heff t)]: (42) (4) Energy per Spin The energy per spin is given by E (t) = h m (t) N

z z

h0 mx (t) cos(!t) + h0 my (t) sin(!t); = hz meqx sin cos f1 cos(heff t)g 2 2 hz meq z fsin + cos cos(heff t)g; 2 2 h0 meq x [fcos + sin cos(heff t)g cos(!t) sin sin(heff t) sin(!t)] cos(!t) h0 meq z [sin cos f1 cos(heff t)g cos(!t) + cos sin(heff t) sin(!t)] cos(!t) +h0 meqx [ fcos2 + sin2 cos(heff t)g sin(!t) sin sin(heff t) cos(!t)] sin(!t) +h0 meqz [ sin cos f1 cos(heff t)g sin(!t) + cos sin(heff t) cos(!t)] sin(!t): (43) VI. DISCUSSION

Let us now examine the implications of our exact expressions given in the previous section via numerical evaluation. Since the y-components of these quantities usually show behaviors that are either phase-shifted or similar to the x-components of those quantities, we shall examine the x- and the z-components only. Moreover,


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Fig. 1. The trajectory of the tip of the magnetization vector at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic eld h0 =hz = 0:01 at angular frequencies ! = (a) 0:01, (b) 1:0 and (c) 1:5 in units of rad/s.


since our noninteracting spins are in the presence of external magnetic elds, the temperature dependences of the nonequilibrium quantities are manifested through only variations in the oscillation amplitudes; thus, the nonequilibrium quantities are just related by a scale factor. Hence, for the sake of saving pages, we shall examine our results at one dimensionless inverse temperature only, i.e., hz = 100. In our numerical evaluations,

we also keep the static magnetic eld along the z-axis large so that the dimensionless rotating magnetic eld h0 =hz 1:0 because this is the case to be realized in magnetic resonance experiments. (1) Magnetization In order to study the time evolution of the magnetization, in Figures 1, 2 and 3, we have plotted the trajectories depicting the time evolution of the tip of the

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magnetization vector at a dimensionless inverse temperature of hz = 100 for dimensionless rotating magnetic elds h0 =hz = 0:01, 0:1 and 1:0, respectively. Here, the angular frequencies are ! = (a) 0:01, (b) 1:0 and (c) 1:5 in units of rad/s. We see here that the magnetic resonance is manifested through the covering of the Bloch sphere [9] by the tip of the magnetization vector at the resonance angular frequency ! !R = hz = 1:0 rad/s and the covering will be denser when the rotating magnetic eld is smaller; thereby, the magnetic resonance is sharper (See Figure 1(b)). Now, in order to see what is going on more closely, we consider the magnetization components. Figures 4 and 5 show the nonequilibrium behavior of the x- and the z-components of the magnetization at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1 and (c) 1:0 and for angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s. We interpret the results given in Figures 4 and 5 as follows: In the presence of a very small transverse rotating eld, i.e., h0 =hz = 0:01 as in Figures 4(a) and 5(a), in the xy-plane, we are almost in a near equilibrium state; thus, the magnetization is almost aligned along the zaxis initially. At a very low angular frequency ! = 0:01 rad/s, little precession is caused. Hence, the x- and the y-components of the magnetization will be almost zero and the z-component of the magnetization will almost maintain a fully aligned state. At the resonance angular frequency ! = !R , due to coherent coupling of motion along the z-axis with that in the xy-plane, suddenly the rotating magnetic eld causes a magnetization wildly precessing about the instantaneous axis along the eective magnetic eld direction; thereby, the z-component magnetization will oscillate slowly with large amplitude and the angular frequency approximately will be equal to the Rabi frequency !Rabi = h0 = 0:01 rad/s while the x- and the y-components of magnetization will oscillate wildly and their envelopes for modulated oscillations will also oscillate approximately with the same Rabi frequency. At a high angular frequency away from resonance, i.e., ! = 1:5 rad/s, due to the incoherent coupling of motion along the z-axis with that in the xy-plane, the magnetization precesses with small radius but rapidly; thus, the x- and the y-components of the magnetization are oscillating rapidly with small amplitudes while the zcomponent of the magnetization remains almost frozen. As we further increase the dimensionless rotating eld to h0 =hz = 0:1, i.e., Figures 4(b) and 5(b), the initial magnetization will be further tilted toward the xy-plane containing the origin from the z-axis and the magnetization precesses around its instantaneous axis along the eective magnetic eld direction with larger radius. Hence, at a very low angular frequency ! = 0:01, the z -component of the magnetization shows very slow, but slightly, enhanced oscillation as a function of time while the x- and the y-components of the magnetization oscillate with increased Rabi frequency !Rabi = h0 = 0:1



rad/s. At the resonance angular frequency !R = hz = 1:0 rad/s, again due to coherent coupling of motion along the z-axis with that in the xy-plane, the magnetization precesses about its instantaneous axis along the eective magnetic eld direction and the more reduced zcomponent magnetization will oscillate slowly with large amplitude and the angular frequency will be approxi-


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Fig. 4. mx vs. time at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1, (c) 1:0 and for angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s.

Fig. 5. mz vs. time at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1 and (c) 1:0 and for angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s.

mately equal to the Rabi frequency !Rabi = h0 = 0:1 rad/s while the x- and the y-components of the magnetization will wildly oscillate and their envelopes for the modulated oscillations also oscillate approximately with the same Rabi frequency. At a high angular frequency away from resonance, i.e., ! = 1:5 rad/s, the magnetization is precessing fast with respect to the instantaneous magnetic eld axis with reduced radius compared to that at the resonance frequency; thereby, the oscillation amplitude for all magnetization components becomes smaller because of incoherent coupling of the magnetization with the rotating eld. Hence, the projection

of the magnetization on the z-axis yields a fast oscillation of the z-component magnetization with smaller amplitudes and the fast oscillations of the x- and the y-components of the magnetization with more reduced amplitudes. When the dimensionless rotating eld is equal to the static eld so that h0 =hz = 1:0, i.e., Figures 4(c) and 5(c), the initial magnetization will be 45 tilted from the z -axis toward the xy-plane containing the origin. Now, the resonance almost vanishes because of the competition between the static eld along the z-axis and the rotating eld in the xy-plane. The magnetization is still

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Fig. 6. < [Sx (t)]2 > vs. time at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1 and (c) 1:0 and for angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s.

Fig. 7. < [Sz (t)]2 > vs. time at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1 and (c) 1:0 and for angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s.

precessing around its instantaneous axis along the eective magnetic eld direction with a radius getting larger as the angular frequency is increased from ! = 0:01 rad/s to 1:5 rad/s. Hence, as we increase the angular frequency, the z-component magnetization oscillates faster with increasing amplitude as a function of time. The x- and the y-components of magnetization also oscillate with increasing amplitudes at the Rabi frequency !Rabi = h0 = 0:1 rad/s. (2) Equal-time Correlation Functions Figures 6 and 7 show the nonequilibrium behaviors of

the x- and the z-components of the equal-time correlation functions at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1 and (c) 1:0 and for angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s, respectively. Since the O(N 2 ) terms dominate over the O(N ) terms, < [Sx (t)]2 > =N 2 < Sx (t) >2 =N 2 = [mx (t)]2 and < [Sz (t)]2 > =N 2 < Sz (t) >2 =N 2 = [mz (t)]2 . Hence, these equal-time correlation functions behave like [mx (t)]2 and [mz (t)]2 so that their interpretations are almost the same as those given for the magnetization ex-


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Fig. 8. < Sx (t)Sx > vs. time at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1 and (c) 1:0 and for angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s.

Fig. 9. < Sz (t)Sz > vs. time at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1,and (c) 1:0 and for angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s.

cept that now they are squared; thus, we shall not elaborate on these anymore. Nevertheless, we should mention here that the equal-time correlation functions are nonstationary because both of them are functions of time. We should contrast this property with that in equilibrium, where they are time independent. We can clearly see this from Figures 6(a) and 7(a) for the dimensionless eld h0 =hz = 0:01 far away from the resonance angular frequency ! = 1:0 rad/s. (3) Time-displaced Correlation Functions Figures 8 and 9 show the nonequilibrium behavior of the x- and the z-components of time-displaced correla-

tion functions at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1 and (c) 1:0 and for angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s, respectively. For the time-displaced correlation functions, Sx (0) and Sz (0) are just numbers, and all the time dependence is coming out of Sx (t) and Sz (t). Hence, < Sx (t)Sx (0) > mx (t)mx (0) and < Sz (t)Sz (0) > mz (t)mz (0) so that they dier from mx (t) and mz (t) by the scale factors mx (0) and mz (0), which can be seen from Figures 8 and 9. We also mention that the non-stationarity property renders the time-displaced correlation functions time-origin sen-

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angular frequency ! = hz = 1:0 rad/s, we expect the oscillating amplitude to grow dramatically at the resonance angular frequency, but to be small away from resonance. As we further increase the dimensionless rotating eld up to h0 =hz = 1:0 rad/s, again the competition between the static eld and the rotating eld diminish the resonance and the oscillation amplitude increases monotonically as we increase the angular frequency, which can be seen in Figure 10. VII. CONCLUSION

In this work, we studied the nonequilibrium statistical mechanics of classical noninteracting spins in a rotating eld with an aim of understanding how the timedependent magnetic eld aects the nonequilibrium behavior. We adopted this simple model because most of the models one can think of are usually unsolvable and we have to take some approximations before we can get the nonequilibrium quantities. Our model shows that even this simple model can exhibit complicated behavior, which can never be found from approximate calculations. More speci cally, our model realized resonance phenomena in a small rotating eld at the resonance angular frequency, but the resonance diminishes in a large rotating eld with increasing oscillation amplitude. REFERENCES

[1] G. F. Mazenko, Equilibrium Statistical Mechanics (Wiley-VCH, Weinheim, 2000); L. P. Kadano, Statistical Physics:

[2] [3]

Fig. 10. < E (t) > vs. time at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1 and (c) 1:0 and angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s.

sitive. (4) Energy per Spin Figure 10 shows the nonequilibrium behavior of the energy per spin at a dimensionless inverse temperature of hz = 100 for dimensionless magnetic elds h0 =hz = (a) 0:01, (b) 0:1 and (c) 1:0 and for angular frequencies ! = 0:01, 1:0 and 1:5 in units of rad/s. The energy per spin contains the eect of all magnetization components. Since all magnetization components exhibit resonance behavior for small rotating elds at the resonance

[4] [5]

[6] [7] [8] [9] [10] [11]

Statics, Dynamics and Renormalization

(World Scienti c, Singapore, 2000). H. S. Song and J. M. Kim, J. Korean Phys. Soc. 49, 1520 (2006). J. Hong and W. Woo, J. Korean Phys. Soc. 49, 2230 (2006). S. K. Oh, J. Korean Phys. Soc. 48, 18 (2006). E. T. Jaynes, Papers on Probability, Statistics and Statistical Physics (Kluwer Academic Publishers, Dordrecht, 1989); Physics and Probability, edited by W. T. Grandy, Jr. and P. W. Milonni (Cambridge University Press, Cambridge, 1993). W. T. Grandy, Jr., Foundations of Statistical Mechanics Vol. II: Nonequilibrium Phenomena (D. Reidel, Dorrecht, 1988). R. Kubo, M. Toda and N. Hashitsume, Statisitical Physics II (Springer-Verlag, Berlin, 1985). G. F. Mazenko, Nonequilibrium Statistical Mechanics (Wiley-VCH, Weinheim, 2006). C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, Berlin, 1978). F. Bloch, Phys. Rev. 70, 460 (1946); R. K. Wangsness and F. Bloch, Phys. Rev. 89, 728 (1953); F. Bloch and I. I. Rabi, Rev. Mod. Phys. 17, 237 (1945). H. C. Torrey, Phys. Rev. 76, 1059 (1949); F. T. Hioe, Phys. Rev. A 30, 2100 (1984); A. V. Alekseev and N. V.

Nonequilibrium Statistical Mechanics of Noninteracting { Suhk Kun Oh Sushilov, Phys. Rev. 46, 351 (1992); H. K. Kim and S. P. Kim, J. Korean Phys. Soc. 48, 119 (2006); C. Brouder, J. Phys. A: Math. Theor. 40, 9455 (2007). [12] I. I. Rabi, N. F. Ramsey and J. Schwinger, Rev. Mod. Phys. 26, 167 (1954).

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[13] J. L. Garcia-Palacios and F. J. Lazaro, Phys. Rev. B 58, 14937 (1998). [14] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1965).