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On the complete aggregation of the Wigner-Lohe model for identical potentials

  • Received: 01 March 2022 Published: 20 May 2022
  • Primary: 82C10, 82C22; Secondary: 35B40, 35Q40

  • We study the collective behaviors of the Wigner-Lohe (WL) model for quantum synchronization in phase space which corresponds to the phase description of the Schrödinger-Lohe (SL) model for quantum synchronization, and it can be formally derived from the SL model via the generalized Wigner transform. For this proposed model, we show that the WL model exhibits asymptotic aggregation estimates so that all the elements in the generalized Wigner distribution matrix tend to a common one. On the other hand, for the global unique solvability, we employ the fixed point argument together with the classical semigroup theory to derive the global unique solvability of mild and classical solutions depending on the regularity of initial data.

    Citation: Seung-Yeal Ha, Gyuyoung Hwang, Dohyun Kim. On the complete aggregation of the Wigner-Lohe model for identical potentials[J]. Networks and Heterogeneous Media, 2022, 17(5): 665-686. doi: 10.3934/nhm.2022022

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  • We study the collective behaviors of the Wigner-Lohe (WL) model for quantum synchronization in phase space which corresponds to the phase description of the Schrödinger-Lohe (SL) model for quantum synchronization, and it can be formally derived from the SL model via the generalized Wigner transform. For this proposed model, we show that the WL model exhibits asymptotic aggregation estimates so that all the elements in the generalized Wigner distribution matrix tend to a common one. On the other hand, for the global unique solvability, we employ the fixed point argument together with the classical semigroup theory to derive the global unique solvability of mild and classical solutions depending on the regularity of initial data.



    Collective behaviors often appear in many classical oscillatory systems [1,4,7,17,18,25,27,29]. Recently, such classical synchronization dynamics has been extended to a quantum regime, and it is called quantum synchronization in literature. It is worthwhile mentioning from [19,20] that quantum synchronization has attracted many researchers in the quantum optics community due to its powerful applications in quantum information and quantum computing [8,14,15,16,21,28,33,34]. Among possible candidates describing quantum synchronization, we are interested in analytical studies on quantum synchronization via Wigner's formalism [30] that was first introduced by Wigner in 1932 in order to find quantum corrections to classical statistical mechanics. For the mathematical properties of the Wigner transform, we refer the reader to [32].

    To set up the stage, we begin with the Schördinger-Lohe (SL) model [19]. Let ψj=ψj(t,x):R+×RdC be the wave function of the quantum system situated at the j-th node whose dynamics is governed by the Cauchy problem to the SL model:

    {itψj=12Δψj+Vjψj+iκ2NNk=1(ψkψj,ψkψj,ψjψj),t>0,xRd,ψj(0,x)=ψ0j(x),j[N]:={1,,N}, (1)

    where Vj=Vj(x):RdR is a time independent potential at the j-th node, κ denotes a (uniform) nonnegative coupling strength between nodes, and , is the standard inner product in L2(Rd).

    Note that the Planck constant is assumed to be unity for simplicity. Like the classical Schrödinger equation, system (1) satisfies L2-conservation of the wave function ψj. We refer the reader to a recent review article [9] for the emergent dynamics of the SL model (1).

    In this paper, we study the emergent dynamics of the Cauchy problem to the WL model with identical potentials:

    {twij+pxwij+Θ[V](wij)=κ2NNk=1{(wkj+wik)(R2d(wik+wkj)dxdp)wij},t>0,(x,p)R2d,wij(0,x,p)=w0ij(x,p),i,j[N], (2)

    subject to initial constraints:

    R2dw0iidxdp=1,|R2dw0ijdxdp1|<1,ij[N]. (3)

    First, we recall the following definition of the emergent dynamics as follows.

    Definition 1.1. [3] System (2) exhibits complete aggregation if relative states tend to zero asymptotically.

    limtwijwmL2(R2d)=0,i,j,,m[N].

    In the sequel, we provide several comments on the Cauchy problem (2)–(3). First, the WL model (2) was first introduced in [3], and a priori asymptotic analysis has been studied only for the two-particle system with N=2. Second, one notices that (2) is equipped with the identical potential V for all i,j. In fact, potentials for the corresponding SL model (1) would be non-identical in general. However, for the simplicity of mathematical representation, identical potentials are considered in both [3] and our analysis of this work. Third, initial data are restricted to a suitable class. For (3)1, it is usually assumed for the classical Wigner equation so that conservation laws hold under the assumption. Precisely, it corresponds to the mass conservation for a classical Schrödinger equation (see Remark 1). Hence, it would be reasonable to employ (3)1. On the other hand, the assumption (3)2 does not appear in the study of the classical Wigner equation consisting of a single equation, whereas (2) contains N2 equations. Later, we will see from Corollary 1 that (3)2 guarantees the uniform L2-boundedness. We refer the reader to [5,6,10,11,13,22,23,26,35] for the Wigner and Wigner-type equations.

    The main results of this paper are two-fold. First, we provide the complete aggregation dynamics of (2) in a priori setting. Under the assumptions (3) on initial data, we can find an invariant set whose center plays the role of an asymptotically stable fixed point (see Lemma 3.2). Then, we obtain the uniform-boundedness of the L2-norm of a solution to the WL model and show that the L2-norms of all relative states tend to zero (see Corollary 1). For details, we refer to Theorem 3.1 in Section 3.

    Second, we provide a global existence theory of (2) combining the classical methods (fixed point theorem and semigroup theory) and exponential aggregation estimates. We highlight that this paper extends the results in [3] where the existence theory was not considered even for N=2, whereas a priori aggregation estimates were established only for N=2. For this, we first define a suitable function space X which is a subset of L2(R2d). Then, we recast the WL model as a first-order matrix-valued PDE on X and apply the fixed point theorem to show that the WL model admits a global mild solution. Furthermore, if more regularity assumptions on initial data are imposed, then we show tha a global classical solution can be obtained from the semigroup approach (see Theorem 4.1).

    The rest of this paper is organized as follows. In Section 2, we introduce generalized Wigner functions and the WL distribution matrix, and study their elementary properties. We also review previous results for the WL model. In Section 3, we provide complete aggregation estimates for the WL model in a priori setting. In Section 4, we show the global existence of mild and classical solutions depending on the regularity of initial data. Finally, Section 5 is devoted to a brief summary of this paper and some remaining issues for a future work. In Appendix A, we summarize classical results on the semigroup theory to be used for the global solvability in Section 4.

    Gallery of Notation: Throughout the paper, as long as there is no confusion, we simply use R2d instead of Rdx×Rdp or R2dx,p. Let f=f(x,p) and g=g(x,p) be two functions in L2(R2d). Then, the standard L2-inner product and the L2-norm are defined by

    f,g:=R2df(x,p)¯g(x,p)dxdp,f:=f,f,

    where ¯g(x,p) is the complex conjugate of g(x,p)C. We set the Fourier transform and its inverse transform as follows:

    (Fϕ)(p):=Rdϕ(x)eixpdx,(F1ϕ)(x):=1(2π)dRdϕ(p)eixpdp.

    For a given real-valued function ψ with two set of variables x,yRd, we define the Fourier transform in y variable as follows:

    (Fypψ)(x,p):=Rdψ(x,y)eiypdy.

    In this section, we introduce the N×N Wigner-Lohe (WL) distribution matrix associated with the SL model (1), and its governing model "the Wigner-Lohe model", and review the emergent behaviors of the 2×2 Wigner-Lohe model in [3].

    In this subsection, we show how the WL distribution matrix can be constructed from the SL model. For this, we first recall the generalized Wigner distribution and the pseudo-differential operator.

    Definition 2.1. [3]

    1. For any two complex-valued wave functions ψ,ϕL2(Rd), the generalized Wigner distribution w[ψ,ϕ] is defined by

    w[ψ,ϕ](x,p):=1(2π)dRdψ(x+y2)¯ϕ(xy2)eipydy,(x,p)R2d, (4)

    where ¯ψ is the complex conjugate of ψ.

    2. For VL(Rd) and wL2(R2d), we define the pseudo-differential operator Θ[V] as

    Θ[V](w)(x,p):=i(2π)dRd[V(x+y2)V(xy2)](Fpyw)(x,y)eipydy=i(2π)dR2d[V(x+y2)V(xy2)]w(x,p)ei(pp)ydpdy.

    Remark 1. Below, we give several comments on the generalized Wigner distribution and the pseudo-differential operator.

    1. The generalized Wigner distribution is complex conjugate symmetric in the sense that

    w[ϕ,ψ](x,p)=1(2π)dRdϕ(x+y2)¯ψ(xy2)eipydy=1(2π)dRd¯ψ(x+y2)ϕ(xy2)eipydybyyy=¯1(2π)dRdψ(x+y2)¯ϕ(xy2)eipydy=¯w[ψ,ϕ](x,p). (5)

    2. For the case ψ=ϕ, two definitions (4) and (5) yield the standard Wigner function [12]:

    w[ψ,ψ](x,p)=1(2π)dRdψ(x+y2)¯ψ(xy2)eipydy.

    Since w[ψ,ψ] coincide with the standard Wigner function, we simply denote

    w[ψ,ψ]=:w[ψ].

    Moreover, one can easily verify that w[ψ] is real-valued.

    3. The p-integral of w[ψ]=w[ψ,ψ] is the modulus square of ψ:

    Rdw[ψ](x,p)dp=|ψ(x)|2.

    Moreover, the (x,p)-integral of w[ψ,ϕ] is the inner product of ψ and ϕ:

    R2dw[ψ,ϕ](x,p)dxdp=1(2π)dR3dϕ(x+y2)¯ψ(xy2)eipydydxdp=Rdϕ(x)¯ψ(x)dx=ϕ,ψ.

    4. Since V is real-valued, one also has

    ¯Θ[V](w)(x,p)=i(2π)dR2d[V(x+y2)V(xy2)]ˉw(x,p)ei(pp)ydpdy=i(2π)dR2d[V(xy2)V(x+y2)]ˉw(x,p)ei(pp)ydpdy=i(2π)dR2d[V(x+y2)V(xy2)]ˉw(x,p)ei(pp)ydpdybyyy=Θ[V](¯w)(x,p).

    In the following lemma, we provide several properties of Θ[V](w) in Definition 2.1.

    Lemma 2.2. For f,gL2(R2d), one has the following relations:

    (i)R2dΘ[V](f)gdxdp=R2dΘ[V](g)fdxdp.(ii)R2dΘ[V](f)dxdp=0.

    Proof. (ⅰ) We use the change of variables:

    (p,p,y)(p,p,y)

    to yield

    R2dΘ[V](f)gdxdp=i(2π)dR4d[V(x+y2)V(xy2)]f(x,p)g(x,p)ei(pp)ydpdydxdp=i(2π)dR4d[V(xy2)V(x+y2)]f(x,p)g(x,p)ei(pp)ydpdydxdp=R2dΘ[V](g)fdxdp.

    (ⅱ) By the definition of Definition 2.1 and Fubini's theorem, we have

    R2dΘ[V](f)dxdp=i(2π)dR4d[V(x+y2)V(xy2)]f(x,p)ei(pp)ydpdydxdp=i(2π)dR3d[V(x+y2)V(xy2)]eipy(Rdf(x,p)eipydp)dydxdp=i(2π)dR3d[V(x+y2)V(xy2)](Fpyf)(x,y)eipydydxdp=iRd[V(x)V(x)](Fp0f)(x,0)dx=0.

    Remark 2. If we set f=u and g=¯u in Lemma 2.2, and recall Remark 2.2(4), then one has

    R2dΘ[V](u)ˉudxdp=R2dΘ[V](ˉu)udxdp=R2d¯Θ[V](u)udxdp=¯R2dΘ[V](u)ˉudxdp.

    This yields

    Re[R2dΘ[V](u)¯udxdp]=0.

    This fact was used in [23] to show the conservation of L2-norm for the solution to the quantum Liouville equation, which can be considered as a special case of the WL model with κ=0.

    For a given ensemble of wave functions {ψj} which is a solution to the SL model (1), we set

    wij:=w[ψi,ψj],wi:=w[ψ]=w[ψi,ψi],i,j[N].

    Then, the evolution of the WL distributions {wij} is given by the following coupled system:

    twij+pxwij=i(2π)dR2d[Vi(x+y2)Vj(xy2)]wij(x,p)ei(pp)ydpdy+κ2NNk=1[(wkj+wik)(R2dwikdxdpR2dwiidxdp+R2dwkjdxdpR2dwjjdxdp)wij]. (6)

    For the detailed derivation of (6), we refer the reader to [3]. Next, we show that system (6) admits conservation laws.

    Lemma 2.3. Let {wij} be a solution to (6) which rapidly decays to zero at infinity. Then, one has

    (i)ddtR2dwii(t,x,p)dxdp=0,t>0,i[N].

    (ii)ddtR2dwij(t,x,p)dxdp=Rd(Vi(x)Vj(x))wij(t,x,0)dx,t>0,i,j[N].

    Proof. (ⅰ) It follows from (6) that

    twii+x(pwii)+Θ[Vi](wii)=κ2NNk=1[(wki+wik)(R2dwikdxdpR2dwiidxdp+R2dwkidxdpR2dwiidxdp)wii]. (7)

    Now, we integrate (7) over (x,p)R2d and then use the zero far field assumption on wij and the second estimate of Lemma 2.3 to see

    ddtR2dwii(t,x,p)dxdp=0.

    (ⅱ) For the second assertion, we follow a similar calculation in (ⅰ).

    Remark 3. (i) Consider the linear Wigner equation:

    tw+pxw+Θ[V](w)=0.

    Then by Lemma 2.3, one has

    ddtR2dw(t,x,p)dxdp=0. (8)

    However, it is worthwhile mentioning that since w(t,x,p) can take negative values, the L1-norm of w would not be conserved in general. In fact, the relation (8) corresponds to the L2-conservation of the Schrödinger equation. If we assume that

    w=w[ψ]

    for a solution ψ to the linear Schrödinger equation, then the following relation holds (see Remark 1 (3)):

    R2dw(t,x,p)dxdp=ψ(t)2L2,t>0.

    Thus, the results in Lemma 2.3 is consistent with the classical theory for the Wigner equation.

    (ii) It follows from Moyal's identity [31] that

    w(t)2=ψ(t)4.

    Hence, the linear Wigner equation enjoys L2-conservation (but not L1-conservation).

    (iii) We have shown in Lemma 2.3 that diagonal elements {wi} satisfy conservation laws. However the off-diagonal elements do not satisfy conservation laws. Hence, conservation laws hold for system (6) with identical potentials.

    From now on, we are concerned with the following special situation:

    Vi=VandR2dw0i(x,p)dxdp=1,i[N]. (9)

    In this case, the (x,p)-integrals of {wij} are constants along the dynamics (6) (see Lemma 2.3):

    R2dwi(t,x,p)dxdp=R2dw0i(x,p)dxdp=1,t0,i[N],R2dwij(t,x,p,)dxdp=R2dw0ij(x,p)dxdp,t0,i,j[N].

    Hence, the Cauchy problem for system (6) with (9) can be further simplified as follows:

    {twij+pxwij+Θ[V](wij)=κ2NNk=1[(wik+wkj)(R2d(wik+wkj)dxdp)wij],t>0,(x,p)R2d,wij(0,x,p)=w0ij(x,p),i,j[N].

    In this subsection, we briefly recall the result from [3] for a two-body system. Extension to the many-body system will be discussed in the following two sections separately. We set

    w+12:=Re[w12],z12(t):=R2dw12(t,x,p)dxdp,R12(t):=Re[z12(t)]. (10)

    Then, (w1,w2,w12) satisfies the Cauchy problem:

    {tw1+pxw1+Θ[V](w1)=κ2(w+12R12w1),t>0,tw2+pxw2+Θ[V](w2)=κ2(w+12R12w2),tw12+pxw12+Θ[V](w12)=κ4(w1+w22z12w12),(w1(0),w2(0),w12(0))=(w01,w02,w012), (11)

    subject to constraints:

    R2dw01dxdp=R2dw02dxdp=1,|R2dw012dxdp|1,R2dw012dxdp1. (12)

    Theorem 2.4. [3] Let (w1,w2,w12) be a solution to (11)(12). Then, we have

    |1z12(t)|

    where is defined in (10).

    Proof. (ⅰ) The first estimate follows from the following ODE:

    This can be integrated explicitly:

    (13)

    where . If we assume that

    (14)

    then the denominator of the right-hand side of (13) can be zero, and hence can blow up in finite-time. Precisely, there exists such that

    In other words, for initial data satisfying (14), would not be bounded. Thus, should be not be small enough to prevent a finite-time blow-up, and such condition is realized as (12). Of course, condition (12) would not be optimal in the sense to guarantee the finite-time blow-up.

    (ⅱ) It is easy to see that satisfies

    (15)

    We multiply (15) by and integrate the resulting relation to obtain

    Then, Grönwall's inequality and the first estimate yield the desired second estimate.

    Before we close this section, we introduce elementary estimates to be used in the following sections.

    Lemma 2.5. Let be a -function satisfying a differential inequality:

    (16)

    Then, the following assertions hold.

    1. If and satisfy

    there exist uniform positive constants and such that

    2. If and satisfy

    there exists a uniform constant such that

    Proof. (ⅰ) By the comparison principle of ODE and method of integrating factor, we have

    Hence, there exist uniform positive constants and such that

    (ⅱ) We multiply (16) with the integrating factor

    to find

    In this section, we present complete aggregation estimates for (2)–(3) in a priori setting. Our first result can be stated as follows.

    Theorem 3.1. Let be a sufficiently smooth solution to . Then, the complete aggregation emerges asymptotically:

    (17)

    Proof. Since the proof is rather lengthy, we introduce a strategy toward the proof. We first claim:

    For this, the key idea is to derive Grönwall's type differential inequality for . To be more specific, we will show that there exist two positive constants and such that

    (18)

    Then, we apply Lemma 2.5 to derive the desired zero convergence for . Finally, the triangle inequality gives the desired result:

    The derivation of (18) will be given in Section 3.2 after some preparatory estimates in Section 3.1.

    In this subsection, we study basic estimates for (2)–(3) that will be used in the derivation of (18). We set

    Then, it follows from Lemma 2.3 that

    (19)

    On the other hand, we integrate (2) with respect to to find the finite-dimensional dynamics for :

    (20)

    Due to (19), it is natural to consider the maximal diameter for the set :

    which is expected to converge to zero under a suitable condition.

    Lemma 3.2. (Existence of a positively invariant set) Let be a solution to satisfying the relation:

    (21)

    Then, one has

    Proof. It follows from (20) that

    (22)

    Then, (22) gives

    (23)

    where . Now, we define a set and its supremum:

    By the assumption on initial data, the set is not empty. We claim:

    Suppose to the contrary, i.e.,

    Then, one has

    (24)

    On the other hand, we observe

    For , let be the extremal indices satisfying

    Hence, (23) yields

    which contradicts (24). Since does not blow up in finite time, one has

    and the set is positively invariant along the flow (20).

    Remark 4. Lemma 3.2 says that if initial data satisfy (21):

    then one has

    Thus, the -integral of is uniformly bounded in time.

    As a direct consequence of Lemma 3.2, we can also show the uniform - boundedness of . For this, we define

    Corollary 1. Let be a solution to with initial data satisfying the relation . Then, the following assertions hold.

    1. The functional decays to zero at least exponentially. In particular, one has

    2. The functional is uniformly bounded in time. Precisely, there exists a uniform positive constant such that

    Proof. (ⅰ) It follows from (20) that

    Then, we find a differential inequality for :

    Lastly, we use initial data (21) together with the above Riccati differential inequality to give the desired result.

    (ⅱ) We multiply with (2) and take real parts for the resulting relation to obtain

    (25)

    Now, we integrate (25) with respect to to find

    (26)

    Below, we present estimates for , respectively.

    Case A.1 (Estimate of ): By integration by parts, we see

    Case A.2 (Estimate of ): It directly follows from Remark 2 that

    Case A.3 (Estimate of ): We use the Cauchy-Schwarz inequality and Corollary 1 to see

    In (26), we collect all the estimates in Case A.1–Case A.3 to derive

    This yields

    (27)

    Since tends to zero exponentially, Lemma 2.5 and differential inequality (27) yield the desired estimate.

    In this subsection, we are ready to provide the proof of Theorem 3.1. First, we claim:

    (28)

    Note that if one verifies (28), then (17) follows from the triangle inequality:

    We consider the difference between and to obtain

    (29)

    Similar to the proof of Corollary 1, we multiply to (29), take real parts and integrate the resulting relation with respect to to obtain

    (30)

    Below, we present estimates of , respectively.

    Case B.1 (Estimates of and ): It follows from the estimates of and in Corollary 1 that

    Case B.2 (Estimate of ): Note that

    In (30), we combine all the estimates Case B.1–Case B.2 to find

    (31)

    If we use Corollary 1 with , then (31) becomes

    (32)

    We sum up (32) with respect to to get

    (33)

    Case C.1 (Estimate of ): we use the Cauchy-Schwarz inequality to see

    (34)

    Case C.2 (Estimate of ): we use Corollary 1 to find

    (35)

    In (33), we combine all the estimates (34) and (35) to derive

    Finally, we use Lemma 2.5 to establish (28). This completes the proof of Theorem 3.1.

    In this section, we show the global existence of a unique mild solution to the Cauchy problem for the WL model (2) following the fixed point approach in [23] where a linear Wigner equation is considered. For this, we define a subset , a norm and a transport operator:

    (36)

    Then, it is easy to check that is a Banach space. In addition, since the transport operator for the linear Wigner equation maps to , it is also useful to define the domain of denoted by :

    For the WL model as a perturbation of the linear Wigner equation, it is strongly believed that is crucial for our analysis (Lemma 4.2). Now, we are ready to provide our second result on the global existence of mild and classical solutions to (2).

    Theorem 4.1. For , the following assertions hold.

    1. If initial data and the potential satisfy

    then there exists a unique mild solution to the Cauchy problem :

    2. If we impose further regularity on initial data and the potential

    then there exists a unique classical solution to the Cauchy problem :

    Proof. Since the proof is rather lengthy, we provide the proofs in Section 4.2 and Section 4.3.

    In this subsection, we follow the same strategy in [23] in which the linear Wigner equation has been treated by means of the semigroup approach. First, we begin with an elementary property of the transport operator in (36) for the WL model.

    Lemma 4.2. Let be a classical solution to . Then, the transport operator satisfies

    In other words, the transport operator for maps to .

    Proof. Since a solution to belongs to in Corollary 1, it suffices to show that

    By straightforward calculations, we observe

    (37)

    Below, we estimate one by one.

    Case C.1 (Estimate of ): we use integration by parts to find

    Hence, we see that vanishes:

    Case C.2 (Estimate of ): since and commute particularly for , we get

    By the skew-Hermitian property of in Remark 1, one has

    Case C.3 (Estimate of ): for the summand in

    In (37), we collect all the estimates in Case C.1–Case C.3 to find

    (38)

    We sum up (38) with respect to and use the Cauchy-Schwarz inequality for the last term to derive

    It follows from Corollary 1 that tends to zero exponentially fast and Lemma 2.5 gives the desired uniform boundedness.

    In this subsection, we show that the Cauchy problem for the WL model admits a unique mild solution.

    First, we rewrite (2) as a matrix form to apply the fixed point theorem. For ,

    (39)

    where is an matrix whose entries are all zero except for -component being 1, is an matrix whose elements in -th row are all one, and is an matrix whose elements in -th column are all one. Here, is understood as an matrix whose -th component is , and the integral in the right-hand side is defined in a similar way. For copies of the function space in (36), we define a set and its norm:

    Then, becomes a Banach space. For the time variable, we use the sup norm. Thus, we consider the Banach space

    equipped with the norm

    Now, we are concerned with the global solvability of (39). Let (to be determined later) and consider a map defined by the following prescription: for each , the function is a solution to the Cauchy problem:

    (40)

    We need to check well-definedness and strict contraction of .

    ● (Well-definedness of ): It suffices to show that for , its image also belongs to . As done in Corollary 1, we multiply with (40), take real parts and integrate the resulting relation with respect to to verify that .

    ● (Strict contraction): for with , our goal is to find a constant such that

    If we set and , then

    For simplicity, we set

    By straightforward calculation,

    (41)

    In addition, we observe

    (42)

    It follows from Corollary 1 that

    Then, (41) and (42) yield

    (43)

    and integrate the relation (43) to find

    Since , we obtain

    If , the map becomes a strict contraction on the closed subset of the (complete) metric space . Hence, has a unique fixed point in for each which gives a unique local solution. Then, it can be globally extended due to uniform estimate or classical way by induction. Precisely, we define by the unique solution to the main equation on with the initial data . Then, we denote

    Hence, becomes the unique global solution to the main equation with the initial data . This completes the proof.

    Next, we are concerned with a global classical solution. In order to apply Theorem A.2(2) in Appendix A for a classical solution to the Cauchy problem (2)–(3), we have to show the continuously differentiability of the coupling term containing .

    For , we introduce an matrix whose -th component is given as

    which is nonlinear with respect to the argument . Since , one can easily verify that maps from to for any . Below, we show that is indeed Lipschitz.

    Lemma 4.3. For , there exists a positive constant that may depend on time such that

    Then, the functional derivative, denoted by , is continuous. Consequently, is Lipschitz from a bounded subset of to .

    Proof. For , we define the Gâteaux derivative (or it is sometimes simply referred as the functional derivative) of at in the direction of that is denoted by :

    At each point , the Gâteaux derivative maps from to . Then by the definition of the Gâteaux derivative, we calculate for and ,

    Since , one finds

    Therefore, we verified that is a bounded linear operator on . Hence, is continuous on . Finally, we recall the Gâteaux mean value theorem in Lemma A.3:

    Here, denotes the operator norm when we regard as a linear operator which maps from to . Since we know that is a bounded linear operator, we find the desired constant:

    This shows that is Lipschitz since and belong to a bounded subset of .

    Now, we are ready to provide the second assertion of Theorem 4.1 by applying semigroup theory.

    Step A (the linear Wigner equation on ): As a first step, we begin with the linear equation in the space instead of :

    (44)

    Since (44) on has been studied in [23], we slightly modify the proof of [23] to show the existence in . In order to use Theorem A.1, we show the term is a a bounded perturbation of the transport operator in . However, when the -norm is considered, Lemma 2.2 gives

    Since is a bounded perturbation of in (see [23,Lemma 1]), we then conclude that is a linear bounded operator defined on . Hence, (44) admits a unique classical solution

    For details, we refer the reader to [23,Theorem 1].

    Step B (the WL model on ): Next, we recall from Lemma 4.2 that is continuously differentiable and then apply Theorem A.2(2) to guarantee that a mild solution obtained from the first assertion of Theorem 4.1 indeed becomes a classical solution to (2)–(3):

    This completes the proof.

    In this paper, we have studied the complete aggregation estimate and the global existence of the Wigner-Lohe(WL) model which describes quantum synchronization in the Wigner picture. By taking the Wigner transform on the Schrödinger-Lohe model with identical potentials, we formally derived the WL model which is an integro-differential equation. Compared to the linear Wigner equation, one of the main difficulty to deal with the WL model lies in the lack of conservation laws. However, fortunately, we can overcome the loss of several conserved quantities via collective dynamics. For the WL model, we first establish complete aggregation estimates that can be achieved with an exponential convergence rate in a priori setting. Next, we show that the WL model admits a unique global mild solution by the standard fixed point theorem and if we impose further regularity on initial data, a unique global classical solution can be obtained by using the semigroup theory. Of course, there are still lots of untouched issues. For instance, we focused on the identical WL model where external one-body potentials are assumed to be the same. Thus, the extension of collective dynamics and global solvability of the WL model with non-identical potentials are left for a future work.

    In this appendix, we briefly summarize several results in [24] on the semigroup theory to show the existence of evolution equations. The first theorem deals with the bounded perturbation of a linear equation.

    Theorem A.1. [24] Let be a Banach space, and let and be operators on such that

    (i) A is the infinitestimal generator of a -semigroup on satisfying

    (ii) is a bounded linear operator on .

    Then, is the infinitestimal generator of a -semigroup on satisfying

    Consider the following abstract Cauchy problem:

    (45)

    In next theorem, we recall the result on the mild and classical solutions of (45).

    Theorem A.2. [24] The following assertions hold.

    1. Let be continuous in on and uniformly Lipschitz continuous (with a Lipschitz constant ) on . If is the infinitestimal generator of a semigroup for on , then for every , the initial value problem has a unique mild solution . Moreover, the mapping is Lipschitz continuous from into .

    2. Let be the infinitestimal generator of a semigroup for on . If is continuously differentiable from into , then the mild solution of with is a classical solution of the initial value problem.

    Finally, we recall Gâteaux's mean value theorem. We denote the directional derivative of at in direction by :

    Lemma A.3. [2,Proposition A.2] Let be a function between Banach spaces and . If is Gâteaux differentiable, then for ,

    where is a bounded linear operator . Here, is the directional derivative of at in direction :



    [1] The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. (2005) 77: 137-185.
    [2]

    B. Andrews and C. Hopper, The Ricci Flow in Riemannian Geometry, Springer Science & Business Media, 2011.

    [3] The Wigner-Lohe model for quantum synchronization and its emergent dynamics. Netw. Hetero. Media (2017) 12: 403-416.
    [4] A shocking display of synchrony. Phys. D (2000) 143: 21-55.
    [5] Some remarks on the Wigner transform and the Wigner-Poisson system. Le Matematiche (1991) 46: 429-438.
    [6] The three-dimensional Wigner-Poisson problem: Existence, uniqueness and approximation. Math. Methods Appl. Sci. (1991) 14: 35-61.
    [7] Biology of synchronous flashing of fireflies. Nature (1966) 211: 562-564.
    [8] Robust quantum gates on neutral atoms with cavity-assisted photon scattering. Phys. Rev. A (2005) 72: 032333.
    [9]

    S.-Y. Ha and D. Kim, Collective dynamics of Lohe type aggregation models, archived as arXiv: 2108.10473.

    [10] Existence, uniqueness and asymptotic behavior of Wigner-Poisson and Vlasov-Poisson systems: A survey. Transport Theory Stat. Phys. (1997) 26: 195-207.
    [11] Global existence, uniqueness and asymptotic behavior of solutions of the Wigner-Poisson and Schrödinger-Poisson systems. Math. Methods Appl. Sci (1994) 17: 349-376.
    [12]

    G. B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies, 122. Princeton University Press, Princeton, NJ, 1989.

    [13] Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. (1997) 50: 323-379.
    [14] Quantum correlations and mutual synchronization. Phys. Rev. A (2012) 85: 052101.
    [15] Quantum stochastic synchronization. Phys. Rev. Lett. (2006) 97: 210601.
    [16] The quantum internet. Nature (2008) 453: 1023-1030.
    [17]

    Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag. Berlin. 1984.

    [18]

    Y. Kuramoto, International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Theoretical Physics, 30, 420, 1975.

    [19] Quantum synchronization over quantum networks. J. Phys. A (2010) 43: 465301.
    [20] Non-Abelian Kuramoto model and synchronization. J. Phys. A (2009) 42: 395101.
    [21] Quantum synchronization effects in intrinsic Josephson junctions. Phys. C (2008) 468: 689-694.
    [22] On the equivalence of the Schrödinger and the quantum Liouville equation. Math. Methods Appl. Sci. (1989) 11: 459-469.
    [23] An analysis of quantum Lioville equation,. Z. Angew. Math. Mech. (1989) 69: 121-127.
    [24]

    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, Berlin, 1983.

    [25]

    C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York, 1975.

    [26] The one-dimensional Wigner-Poisson problem and a relation to the Schrödinger-Poisson problem. SIAM J. Math. Anal. (1991) 22: 957-972.
    [27] From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Phys. D (2000) 143: 1-20.
    [28] Superinsulator and quantum synchronization. Nature (2008) 452: 613-616.
    [29] Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. (1967) 16: 15-42.
    [30] On the quantum correction for thermodynamic equilibrium. Phys. Rev. (1932) 40: 749-759.
    [31]

    M. W. Wong, Weyl Transforms, Springer, New York, 1998.

    [32]

    P. Zhang, Wigner Measure and Semiclassical Limits of Nonlinear Schödinger Equations, Courant Lecture Notes in Mathematics, vol. 17, 2008.

    [33] Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator. Phys. Rev. B. (2009) 80: 014519.
    [34] Quantum synchronization. Eur. Phys. J. D. (2006) 38: 375-379.
    [35] The Wigner transform and the Wigner-Poisson system. Transp. Theory Stat. Phys. (1993) 22: 459-484.
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    1. Seung-Yeal Ha, Gyuyoung Hwang, Dohyun Kim, On the semiclassical limit of the Schrödinger-Lohe model and concentration estimates, 2024, 65, 0022-2488, 10.1063/5.0194571
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