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Strange Scaling and Temporal Evolution of Finite-Size Fluctuation in Thermal Equilibrium

We numerically exhibit strange scaling and temporal evolution of finite-size fluctuation in thermal equilibrium of a simple long-range interacting system. These phenomena are explained from the view point of existence of the Casimirs and their nonexactness in finite-size systems, where the Casimirs are invariants in the Vlasov dynamics describing the long-range systems in the limit of large population. This explanation expects appearance of the reported phenomena in a wide class of isolated long-range systems. The scaling theory is also discussed as an application of the strange scaling.

statistical mechanics gives γ ± = 1 and δ = 3. The former non-classical critical exponents are again explained by the Casimir constraints, which suppress the response.
We note that the strange critical exponents for the response are obtained not in the forced thermal equilibrium but in QSSs. However, if the fluctuation-response relation holds, then anomalous fluctuation appears in non-forced thermal equilibrium. Similarly, temporal evolution of the finite-size fluctuation can be expected since the strange values of γ − and δ comes from the Casimir constraints, but the finite-size effect releases the system from the constraints as sketched in the relaxation process.
To examine the above scenario, we perform direct Nbody simulations in the HMF model. We remark that validity of the fluctuation-response relation is not obvious since the response theories [21][22][23][24][25] are based on the Vlasov dynamics in which the limit of large population is taken. We, therefore, first confirm the fluctuation-response relation for off-critical points, and report the strange scaling at the critical point corresponding to the non-classical critical exponent δ = 3/2. Then, we show the temporal evolution of finite-size fluctuation in thermal equilibrium. We underline that, since the scenario is based on existence of the Casimirs, the reported phenomena are not limited in the HMF model, but can be expected in generic long-range systems. In addition, we discuss that the strange scaling conjectures values of critical exponents for the correlation length with the aid of the scaling theory [29][30][31].
The HMF model is a paradigmatic long-range system and is expressed by the Hamiltonian where m j = (cos q j , sin q j ) is a XY spin and h = (h x , h y ) is the external force. Each particle is confined on the unit circle, and the phase of j th particle is q j ∈ (−π, π] and p j ∈ R the conjugate momentum. The factor 1/N in the potential term is added to ensure extensivity of energy. The magnetization (order parameter) vector of the HMF model is defined by the arithmetic mean of m j as The HMF model has the second order phase transition between the disordered phase (T > T c ) and the ordered phase (T < T c ) with the critical temperature T c = 1/2 [28]. We denote the canonical average of M as 〈M 〉, and define the zero-field susceptibility tensor χ = (χ ab ) by statistical mechanics as It is straightforward to prove the relation for h = 0 from the explicit expression of 〈M 〉 and the definition of χ. We note 〈M〉 = 0 from symmetry of the system. Further derivations give See Refs. [31,32] for instance. The magnetization vector fluctuates around M = 0 in the disordered phase. However, in the ordered phase, the Goldstone mode appears, and the zero-field susceptibility diverges for this direction [22,33]. We, therefore in the latter, concentrate on fluctuation for the radius direction. Consequently, we compute the finite-size fluctuation by the quantity and the fluctuation-response relation is written in the form from the rotational symmetry, where the factor ǫ comes from dimensionality of the fluctuation of M . We numerically examine the fluctuation-response relation (7) and the fluctuation at the critical point (5) by direct N -body simulations. We numerically integrate the canonical equations of motion associated with the Hamiltonian (1) by using the fourth-order symplectic integrator [34] with the fixed time step ∆t = 0.1. The initial values of N pairs of (q j , p j ) are randomly picked up from the equilibrium one-particle distribution function f eq (q, p) = and the zero-field susceptibility ǫχ V xx obtained in the Vlasov dynamics, whose level is reported by the black solid lines. The dot-dashed gray and the dashed light-blue lines in the ordered phase (T < T c = 1/2) are isothermal and isoentropic susceptibilities [26], while they coincide with the Vlasov one in the disordered phase. Points are averages over 100 realizations. Averaging time t av for one realization is t av = 100. N = 10 2 (red triangles), 10 3 (blue diamonds), 10 4 (magenta inverse-triangles) and for instance. We further take the average of V M (N ) over 100 realizations of initial states, but denote it by V M (N ) for simplicity.
The finite-size fluctuation NV M /T is compared with the zero-field susceptibility χ V xx obtained theoretically [22] from the Vlasov dynamics in Fig.1. They are in good agreement in the ordered phase and for large T in the disordered phase. Even around the critical point of the disordered phase, we can find tendency of convergence of the fluctuation to the Vlasov susceptibility level as N increases. Moreover, for the reduced temperature τ = |T − T c |/T c , the scaling of NV M /T ∝ τ −1/4 is found in the ordered phase corresponding to the non-classical exponent γ − = 1/4 defined by χ V xx ∝ τ −γ − . Thus, we conclude that the fluctuationresponse relation holds even under the Casimir constraints and the finite-size effects.
For the strange scaling at the critical point, we introduce the Landau's pseudo free energy per one particle (9) In this framework realized M is derived from the equation  we have the scaling of V M (N ) ∝ N −2/(δ+1) at the critical point, and the classical value of δ = 3 gives the scaling of (5). However, the Vlasov dynamics gives δ = 3/2 [23], and the scaling V M (N ) ∝ N −4/5 is expected accordingly. This expectation is confirmed in Fig.2, and holds even for N ≃ 100 with a short time averaging time t av = 100. We remark that the scaling tends to approach to the classical one, N −1/2 , as increasing t av .
As found in Fig.1, there is a gap between the Vlasov and the isoentropic susceptibilities, and hence one may expect that the fluctuation, NV M (N )/T , temporally evolves from the former level to the latter. This expectation is confirmed in Fig.3 for T = 0.45 and 0.40 by varying the averaging time t av . The initial increase for t av 10 is a natural consequence from the definition of V M as the time average, and hence we omit it. The fluctuation is once trapped at the Vlasov level around t av = 10 − 1000 for N = 1000, and goes towards the isoentropic level.
A qualitative change of dynamics between a short-and a long-time intervals is also captured in Fig.4 by computing power spectra of M 2 (t ) with N = 1000. The power spectra, each of which is the average over 100 realizations, are divided into the short-time interval corresponding to f 0.01 and the long-time one having algebraic dampings as f −a with a from 1.0 to 1.4.
Putting all together, we can now make a scenario of the finite-size fluctuation in the long-range systems. In the limit N → ∞, the infinite Casimirs divide the phase space into their level sets, and any initial state belongs to one of them and can not escape from the level set. When N is finite, invariance of the Casimirs is no longer exact but approxi- mate. Nevertheless, in the short-time interval, the system fluctuates along the approximate level set, which induces the strange scaling of V M ∼ N −4/5 at the critical point. As time goes on, the system is released from the approximate constraints, and the classical scaling V M ∼ N −1/2 recovers in the long-time interval. Temporal evolution of the finitesize fluctuation can be understood as the releasing process from the iso-Casimir contour. An important remark is that any initial states must evolve under the approximate but unavoidable Casimir constraints even in thermal equilibrium.
We further investigate the finite-size fluctuation from the view point of the scaling theory [29]. One hypothesis of the scaling theory is that the finite-size scalings are controlled by the dimensionless quantity ξ/L, where ξ and L are the correlation length and the system length respectively. The infinite-range models, including the HMF model, have no concept of both lengths, but a scaling theory has been proposed for such models by replacing ξ/L with N c /N [30,31], where N c is the coherent number of particles. It is supposed that N c ∝ ξ d u ∝ τ −d u ν with d u the upper critical dimensionality and ν defined by ξ ∝ τ −ν for the mean-field universality class. Thus, ν * = d u ν plays the role of ν, and the scaling function for χ xx , denoted by F χ , can be introduced as Combining the above expression with the fluctuationresponse relation (7), we have The scaling (11) is examined in Fig.5(a) for the disordered phase, and the scaling function F χ can be observed. In the ordered phase, the susceptibility χ xx scales as χ xx ∝ τ −γ − but this scaling is not excellent even for rather small τ, and observation of the scaling function is hard accordingly. The susceptibility is represented as χ xx = (1 − D)/D, and D shows a beautiful scaling as shown in the inset of Fig.5(b). We, therefore, examine the scaling for D, which suggests The scaling function F D is clearly observed in Fig.5(b), and hence we may also expect the scaling function F χ in the ordered phase for small values of τ.
Summarizing, we investigated the finite-size fluctuation in thermal equilibrium of the HMF model. We have confirmed that the fluctuation-response relation holds even under the Casimir constraints and finite-size effects. This relation induces the strange scaling of V M (N ) ∝ N −4/5 at the critical point instead of N −1/2 predicted by statistical mechanics. Further, we reported temporal evolution of the finite-size fluctuation. These phenomena are explained by existence of the Casimirs, which are approximate invariants in finite-size long-range systems. Therefore, these phenomena can be expected in generic systems. In addition, the strange scaling with the scaling theory conjectures critical exponents for the correlation length.
We end this article by giving five remarks. First, the linear and the nonlinear response theories are simply reproduced [24] by the so-called rearrangement formula [25]. It is worth extending the powerful formula for finite-size systems. Second, the evolution of finite-size fluctuation is not expected in the disordered phase, since there is no gap between the Vlasov susceptibility and the isoentropic one. Third, nonexactness of the Casimir invariants is a crucial point for the evolution. Exact invariants of the translational and the angular momenta break the equipartition of kinetic energy in small clusters, but this phenomenon is understood by inputting the invariants into statistical mechanics [36]. Fourth, approximate constraints can be found also in spring-chain systems. Hard springs play the role of approximate constraints, lengths of bonds, and slow relaxation to equipartition is observed [37]. The linked particles can be considered as models of molecules for instance, thus it might be interesting to investigate the finite-size fluctuation in such models. Finally, the strange scaling is observed for a rather small N , say N ≃ 100, when the averaging time is short. This observation suggests that we need to consider the wreck of Casimirs even in small systems, and might give a new perspective to understand more realistic systems.
The author thanks K. A. Takeuchi for suggesting to study the finite-size fluctuation. He also acknowledges the support of JSPS KAKENHI Grant Number 23560069.