First-Order Phase Transition by XY Model of Particle Dynamics
FFirst-Order Phase Transition by XY-Model of Particle Dynamics
Amir H. Fatollahi
Department of Physics, Alzahra University,P. O. Box 19938, Tehran 91167, Iran [email protected]
Abstract
A gas-liquid type of phase transition is found based on the particle dynamics on radius- R circle in which the coordinate appears as the angle-variable of 1D XY-model. Due tothe specific appearance of compact-space radius (volume) in the present interpretationof XY-model, the ground-state develops a minimum at some critical radius, leadingto the multi-valued Gibbs energy similar to systems with first-order phase transition.
Keywords: classical spin models, liquid-vapor transitionsPACS No.: 75.10.Hk, 64.70.F- a r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Introduction
The absence of phase transition in one-dimensional models of magnetic systems is com-monly considered as a special case of the so-called van Hove’s theorem [1]. However, adetailed examination of conditions by the theorem shows that even 1D models may exhibitphase transitions [2].In the present note a model is considered for particle dynamics based on the 1D XY-model of magnetic systems. The model was initially considered in [3], with emphasize onthe phase structure due to the model’s defining-parameter. In the present work the mainconcern is the first-order phase transition based on the
P V equation-of-state by the model,qualitatively similar to the gas-liquid phase transition for systems with for example Vander Waals equation-of-state [4]. In the proposed dynamics, the coordinates are assumedto be compact variables of radius- R circles, appearing as angle-variables of 1D XY-modelsliving on the particle’s discrete worldline. The present approach to particle dynamics issimilar to the one in lattice formulation of gauge theories, in which the gauge variablesappear as compact angle-variables living on a discrete lattice. As will be discussed indetail, due to the specific appearance of the radius in the formulation, the ground-stateenergy develops a minimum at some critical radius, leading to the multi-valued Gibbsenergy quite similar to the systems with gas-liquid first-order phase transition.The organization of the rest of the paper is as follows. In Sec. 2 the model for particledynamics as well as its exact spectrum and the emergence of minimum in the ground-stateare presented. In Sec. 3 the thermodynamics and the phase structure by the model arepresented. It is discussed how the minimum in the ground-state leads to the P V and GP -diagrams similar to those of gas-liquid systems. In Sec. 4 a detailed comparison is madebetween the present and the magnetic interpretations of the XY-model. In particular,the distinguished role by the compact space radius in the announced first-order phasetransition is clarified. Sec. 5 is devoted to concluding remarks and possible extensions ofthe present model. In this section the model and its exact spectrum are presented. However, it is usefulto review the basic elements of the thermodynamics by the ordinary dynamics. It isknown that the ordinary dynamics of free particles does not lead to a phase transition.In particular, the one-particle partition function at temperature T = β − for a particle ofmass m in a d -dimensional box of volume V = L d is given by [5] Z ( β, V ) = (cid:16) m π a (cid:17) d/ (cid:90) L/ − L/ d (cid:89) i =1 N − (cid:89) n =0 dx in e S (1)2n which S is the imaginary-time action in the time-sliced form S = − m a d (cid:88) i =1 N − (cid:88) n =0 ( x in +1 − x in ) (2)with a = β/N as the tiny time-slice parameter. The representation (1) is to be supple-mented by the periodic condition x i = x iN (in the continuous-time form x i (0) = x i ( β )).In the L → ∞ limit (1) reduces to the well-known expression [4] Z ( β, V ) = (2 πm/β ) d/ V (3)By means of the free-energy A = − T ln Z , the equation-of-state P = − ( ∂A/∂V ) T leadsto P = T /V , by which one expects the thermodynamics of a single-phase ideal gas.In the present work the particle dynamics based on the 1D XY-model is again intro-duced by the imaginary-time action, in which instead of the action (2), we consider (inunits (cid:126) = c = 1) S = m R a d (cid:88) i =1 (cid:88) n (cid:18) cos x in +1 − x in R − (cid:19) (4)in which “ a ” appears as the spacing parameter on the discrete worldline. The coordinate x i is treated as compact angle-variable for which we assume − πR ≤ x i ≤ πR (5)As mentioned earlier, the treatment of coordinates as angle-variables may be consideredas the continuation of the agenda originated by lattice formulation of gauge theories, inwhich the gauge fields act as angle variables [6, 7]. The action (4) is in fact the sum of d copies of 1D XY-model, for which it is known there is no phase transition as a magneticsystem [8]. As it will be clarified later, it is the specific appearance of the volume ofcompact space in the present interpretation, namely the presence of radius R both insideand in front of the cosine functions in (4), which leads to the announced phase transition.The action (4) reduces to the ordinary one (2) in the limit x i /R (cid:28)
1. Using thetransfer-matrix method one can obtain the energy spectrum by the imaginary-time (Eu-clidean) action. The element of transfer-matrix (cid:98) V between two adjacent times n and n + 1is given in terms of the imaginary-time action [5] (cid:104) (cid:126)x n +1 | (cid:98) V | (cid:126)x n (cid:105) = (cid:16) m πa (cid:17) d/ exp (cid:34) mR a d (cid:88) i =1 (cid:18) cos x in +1 − x in R − (cid:19)(cid:35) (6)In the Fourier basis (cid:104) (cid:126)x | (cid:126)s (cid:105) = exp(i (cid:126)s · (cid:126)x/R ) / (2 πR ) d/ , it is easy to see that the above transfer-matrix is diagonal [3, 8]. Using the relation (cid:98) V = exp( − a (cid:98) H ) with (cid:98) H as the Hamiltonian,one finds the exact energy spectrum [3, 8] E s , ··· ,s d ( R ) = − a d (cid:88) i =1 ln (cid:34)(cid:114) πmR a e − mR /a I s i ( mR /a ) (cid:35) (7)3 .5 1.0 1.5 2.0 2.5 3.0 R E E E E R * Figure 1:
The plots of three lowest energies in units m = a = 1 for d = 2 case. The ground-state’sminimum is at R (cid:63) (cid:39) . in which s i ’s are integers, and I s is the modified Bessel function. In fact the expression forthe spectrum of XY-model of magnetic systems is essentially as above [8], except for theextra square-root (cid:112) πmR /a , by which a key minimum is developed in the ground-state E , ··· , at radius (see Fig. 1) R (cid:63) = 0 . (cid:112) a/m (8)The minimum in the ground-state is an indication that the system exhibits a first-orderphase transition, as we will see later.The limiting behaviors of the spectrum can be obtained. At the extreme large radiuslimit mR /a (cid:29)
1, using I s ( α ) (cid:39) e α √ πα exp( − s / α ) for α (cid:29) s , one finds the almostcontinuous spectrum E s , ··· ,s d (cid:39) mR d (cid:88) i =1 s i (9)as the ordinary kinetic energy of a free particle moving with momentum p i = s i /R in thecompact direction with radius R . In the small radius limit mR /a (cid:28)
1, using I s ( α ) (cid:39) ( α/ /s ! for α (cid:28)
1, we find for the discrete spectrum E s , ··· ,s d (cid:39) a ln (cid:18) mR a (cid:19) d (cid:88) i =1 ( s i + 12 ) (10)So the continuous spectrum in large radius limit approaches the discrete one in the smallradius limit. The partition function may be evaluated either by the definition Z ( β, R ) := (cid:88) { s i } e − β E { si } ( R ) (11)4
000 2000 3000 4000 5000 6000 V - - - - - - A T = T = T c = T = Figure 2:
The isothermal AV -diagrams for d = 3 (temperature units a − ). The diagrams with T < T c = 0 . or by means of the representation similar to (1) [5] Z ( β, R ) = Tr ˆ V β = (cid:16) m πa (cid:17) d/ (cid:90) πR − πR (cid:89) i, n dx in exp (cid:34) mR a d (cid:88) i =1 β − (cid:88) n =0 (cid:18) cos x in +1 − x in R − (cid:19)(cid:35) (12)supplemented by the periodic condition x = x β . In the present case the equivalence of(11) and (12) is checked numerically. By either (11) or (12) it is easy to see that Z = ( Z d =1 ) d (13)simply because the action (4) is fully separable in x i ’s. By means of the free-energy A = − T ln Z , the partition function (11) can be used to study the phase structure bythe model. Hereafter we work with the choice m = a = 1, and consider the case d = 3,in which the volume and radius are related as V = (2 π R ) or R = V / / π . As theconsequence of the mentioned minimum in the ground-state at radius R (cid:63) , at sufficientlylow temperatures where the ground-state has considerable contribution to Z , there arepoints with common tangents (slopes) in the isothermal AV -diagrams. In Fig. 2 theisothermal AV -diagrams clearly confirm this expectation below the critical temperature T c = 0 . P = − ( ∂A/∂V ) T , in the isothermal P V -diagramsthere should be different volumes with equal pressure, leading to parts with positive slope ∂P/∂V >
0. The positive slope in
P V -diagram would mean a negative compressibility,leading to the mechanical instability of the system [4]. The mentioned behaviors aresimilar to those by the van der Waals equation-of-state, commonly used to describe thesystems with gas-liquid transition [4, 9]. In the real situation, the system has a constantpressure during the gas-liquid transition, the so-called vapor-pressure [4, 9]. To reflectthe real situation, the
P V -diagrams are to be modified based on the so-called Maxwellconstruction [4, 9], by which the constant pressure part finds a thermodynamical basis.5
000 2000 3000 4000 V - P T = T = T c = T = Figure 3:
The isothermal
P V -diagrams at different temperatures. The dashed parts represent thediagrams before the modification by the Maxwell construction. - P - - - - - - G T = . T = . T c = . T = . Figure 4:
The isothermal GP -diagrams. The dashed lines represent the parts that are not beingfollowed by system, leading to jump in slope at cusp. To maintain the stability of system at equilibrium, a proper treatment of the part withpositive ∂P/∂V is essential.In Fig. 3 samples of
P V -diagrams by the present model are plotted, in which bothunmodified paths (dashed lines) and modified ones are present. The Maxwell constructionand the corresponding phase structure are best described based on the Gibbs energy G = A + P V . As the consequence of the mentioned behavior of free-energy, the isothermal GP -diagrams develop cusps below T c [4, 9], after which G is multi-valued for some pressures.For the present model plots of the isothermal GP -diagrams are presented in Fig. 4, inwhich the expected cusps are evident. As for states with equal temperature and pressure,the state with lower G is selected by the system [4], the parts beyond the cusps are notfollowed by the system. Instead, as volume is changed the constant pressure at cusp is holdduring a phase transition. In Fig. 4 the dots at cusps indicate the constant vapor-pressure6igure 5: The
P V T -surface of states for d = 3. The thick curve is at critical temperature T c = 0 . P c = 0 . V c = 409 . values, and each dashed part in Fig. 3 corresponds to a similar part in Fig. 4, both notbeing followed by the system.As at the cusp there is a jump in the first derivative ∂G/∂P , the corresponding transi-tion is categorized as a first-order phase transition [4, 9]. The P V T -surface for the presentmodel is presented in Fig. 5, which is quite reminiscent to the one of a system with thegas-liquid phase transition [4].
The 1D XY-model of magnetic systems is well known to exhibit no phase transition. As inthe present interpretation the system is reduced to d copies of 1D model, it is necessary tounderstand the origin of the phase transition by the present interpretation of the model.The 1D XY-model of magnetic systems is given by the Hamiltonian H Mag = − J (cid:88) n (cid:0) cos( θ n +1 − θ n ) − (cid:1) (14)with J as the coupling constant, and θ n as the orientation of n -th classical spin on the 1Dchain. The partition function for the magnetic system at temperature T = 1 /β is thengiven by [8] Z Mag = (cid:90) π − π (cid:89) n dθ n exp (cid:104) β J (cid:88) n (cid:0) cos( θ n +1 − θ n ) − (cid:1)(cid:105) (15)Now the key point is, in the magnetic interpretation of the model the coupling J is absentinside the cosine functions of (14) and (15), in contrast to the situation in the particle7ynamics interpretation, in which the radius R appears both in front and inside the cosinefunctions of (4) and (12). The way of appearance of radius R makes the fundamentaldifference between two interpretations of the 1D XY-model. In the particle dynamicsinterpretation one may get rid of R inside the cosine function by defining the angle variables θ n = x n /R with − π ≤ θ n ≤ π . However, this does not remove R ’s completely, as theywould rise in the integral-measure of (12), explicitly as Z d =1 = (cid:16) m πa (cid:17) d/ (cid:90) π − π (cid:89) n (cid:0) R dθ n (cid:1) exp (cid:34) mR a β − (cid:88) n =0 (cid:0) cos( θ n +1 − θ n ) − (cid:1)(cid:35) (16)by which the different appearance of R in comparison with J in (15) is observed. Con-cerning the energy spectrum, as mentioned earlier it is easy to check that the presence of (cid:112) mR /a in energy (7) is responsible for the minimum in the ground-state. This extraappearance of R does not have an analog for J in the spectrum of 1D magnetic system [8].Now again, by the new variables θ n = x n /R with the Fourier basis (cid:104) θ | s (cid:105) = exp(i s θ ) / √ π ,using the relation | x (cid:105) = | θ (cid:105) / √ R the radius R comes back as a pre-factor for transfer-matrix(6). In summary, in the particle dynamics interpretation of XY-model, in contrast to the1D magnetic system with no phase transition, the specific appearance of compact spaceradius (volume) in the model is the origin of the phase transition. In this note a particle dynamics interpretation of the XY-model is considered, in whichthe coordinates are assumed to be compact variables, appearing as angle-variables of the1D XY-model. The present interpretation may be considered as a continuation of theagenda leading to the lattice formulation of gauge theories, in which the gauge fields areassumed to be compact angle-variables [6, 7]. Accordingly, this leads to the replacementof the action of the quadratic form (cid:82) F µν by the cosine form (cid:80) P (1 − cos Φ P ), with Φ P as the field-flux inside the plaquette P . In the same way, here the quadratic form (2) foraction of a particle is replaced with (4), leading to the mentioned phase structure.In the present interpretation it is clarified that the specific treatment of the dynamicson compact space in the 1D XY-model leads to a first order phase transition, similar tothe P V -diagrams of gas-liquid systems. In fact there are examples originated from the2D magnetic systems that may exhibit a phase transition of the first-order. In [10, 11]a power-form of the interaction-terms in 2D XY and Heisenberg models are considered,leading to a first-order phase transition for sufficiently large power being used. In thepresent case, however, the first-order phase transition is exhibited by the same power-onecosine-term of 1D XY-model due to the different interpretation.Apart from the theoretical aspects and implications, one may try to find some appli-cations for the present interpretation, for which the above-mentioned relation with lattice8auge theory may come useful. It is known that the lattice formulation of gauge theo-ries approach the ordinary formulation at weak coupling limit g (cid:28)
1, and the behaviorat strong coupling limit is expected to be described by the lattice formulation. In thepresented particle dynamics based on the angle-variables also the ordinary formulation isrecovered at large radius limit R → ∞ by (8), and the possible relevance of the formula-tion would appear at the small- R limit. This leads to a possible application of the presentmodel in the cosmological context. In the present cosmological models it is assumed thatthe matter ingredients obey the ordinary dynamics with the known thermodynamics. Itis of interest to see how the dynamics and phase structure by the present model affectsthe different stages of Universe during the expansion, through which the system is drivenfrom extremely small sizes to its present extent.The extensions of the present model to magnetic systems other than the XY-model,such as Potts model with Z n -valued spins, can be of interest. In the present work thesystem obeyed the Maxwell-Boltzmann statistics. The extensions to Fermi-Dirac andBose-Einstein statistics may lead to interesting features. Acknowledgement : This work is supported by the Research Council of AlzahraUniversity.
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