A New Quenched XY model with Nonusual "Exotic" Interactions
AA New Quenched
X Y model with Nonusual ”Exotic” Interactions
Anderson A. Ferreira Departamento de F´ısica, Universidade Federal de S˜ao Paulo, 96010-900, S˜ao Paulo, Brazil
Human beings live in a networked world in which information spreads very fastthanks to the advances in technology. In the decision processes or opinion formationthere are different ideas of what is collectively good but they tend to go against theself interest of a large amount of agents. Here we show that the associated stochasticoperator ( (cid:100) W ℵ ρ, ∆ ,r ) proposed in [43] for describe phenomena, does not belong to aCP (Contact Processes) universality class [35]. However, its mathematical structurecorresponds to a new Exotic quantum XY model, but unprecedently, their param-eters is a ”function” of the interaction between the local sities ( i − , i, i + 1), andwith the impurity present at the same site i . Introduction
Failure in cooperating can threaten existence itself. Conflicts and issues such as wars,corruption, loss of liberty and tyrannies, environmental degradation, deforestation, amongothers pose great problems and are a testament to humanities inability of cooperate in asuitable level. These examples show that we still do not have a complete understanding ofthe mechanism which drives the collective toward to a common goal and hence to avoid thetragedy of the commons [1, 2]. Despite that, altruism, cooperation and moral norms stillbeing improved to outcompete behaviors of free riders, selfishness and immoral [3, 4].Living organisms and human beings are characterized by autonomy. However, they tendto be prone to selfishness, a bias that may bring harsh damage to their survival as well as theenvironment, maybe due to ambitions and potential short-sightedness. The assumption thatliving organisms are selfish has been accepted by many branches of contemporary science.For example, the inclusive fitness theory, in ecology, in which egoism has biological roots [5].A similar idea arises in neoclassical economic theory, which hypothesized that all choices, nomatter if altruistic or self-destructive, are designed to maximize personal utility [6]. Thus,decisions are motivate by self-interest.One of the most interesting issues to be addressed in the context of the present study is a r X i v : . [ phy s i c s . s o c - ph ] M a y morality. The human free will combined with its selfish/altruist nature may create a plethoraof different patterns over several types of social systems. How does morality emerge? Thisquestion probably does not have a simple and unique answer. Many thinkers from ancienttimes to today, tries to unravel this phenomenon. As a legacy we have a body of theoriesthat seek to understand and explain the emergence of morality in different societies. ThomasHobbes [7, 8] was one of the first modern philosopher to offer a naturalist principle toethics. In his theory, ethics emerge when people understand the necessary conditions tolive well. According to Hobbes, these conditions are defined by imposition of equality ofrights, by means of an absolute Sovereign, due to the necessity of self preservation and byestablishing deals among individuals. Latter on, Rousseau proposed that life in communitycan lead to the loss of individual freedom since the subjects must fulfill a social contractexpressed through laws and institutions [9]. Unlike the philosophers who attributed toreason the capacity to conceive morality, Durkheim understood it as a result of a set ofsocial interactions and culture elaborated throughout history [10, 11]. But these are part ofa small selection of seminal works about a theme hallmarked by an intriguing and challengescientific problem. This debate continues in different areas such as Psychology [12, 13],Political Science, Philosophy, Antropology [14], Education, Economics and Ecology [15, 16].To further advance the long discussion on morality or cooperation, we need to understandsome specific mechanisms of social interaction in various scenarios with different individualdegrees of freedom, effective individual choices, and consider that these choices are influ-enced locally by peers’ s opinions [17–19]. Different levels of freedom (free choice), control(supervision) and social dynamics impact the individual capacity to fulfillment of the so-cial contract and hence should lead to different degrees of morality or cooperation at thecollective level.It is a well-known fact that a system of interacting linked individuals can work togetherto reach a collective goal. Understanding how decentralized actions can lead to these resultshas been a topic of study in the literature for decades. The focus of our study is the roleof a master node, connected to some members of a society, may drive the pursued idealby collectiveness. The topology formed by a master node connected to a network mayrepresent many situations in social systems: law enforcement and citizens [20–22]; moraland community [13, 17–19]; beliefs and member of churchs [23–25]; cooperation and egoism[26, 27]; tax evasion and fiscal country, among others. In all examples, individuals do notshare the same goals, due to the incentives in acting against the common good.We approach this issue using a stochastic quenched disorder model to study the consensusformation [28, 29]. In this model, individuals are autonomous to make decisions based ontheir own opinions or let decisions be influenced by a local social group or/and by thepresence of a norm (master) that reinforces preferential behaviors. The individual decisionsare binaries (0 or 1) and the collective decision is the average collective decision.This model does not belong to a class of nonequilibrium systems [30–32]. We foundabsorbing states phase transitions with respect to three distinct order parameter [33–36].From a statistical mechanics point of view, phase transition in nonequilibrium sytems arestudied by fundamental concept as scaling and universality class [37–40], which may reservesome unexpected results [41, 42].The remainder of the paper is organized as it follows: in Sec. I we define the model andintroduce the general notation. In Sec. II we change the notations to construct the newoperator. For instance, we use the special parametrization [43] and the Schutz’s prescription[44] to write a XY model with a new exotic topological interaction. Finally, in Sec. IVcontains conclusion and an outlook on future work. I. THE MODEL
In Figure 1 we illustrate our model. It consists of a ring formed by nodes with periodicboundary conditions. Initially, each node represents particles or agents which may assumetwo different states s = 0 , w s . If , this means that there may be an intrinsictendency or preference of particles for a determined state s . So, in principle, this is a particleproperty. They interact with first neighbors (on the left and on the right). Moreover, weintroduce a master node illustrated by a large sphere on the top of the ring in Figure 1.The master node connects with particles located in the ring with probability in the initialtime (quenched disorder). The interaction strength between master and connected agents isdenoted by r . The general configuration of the system is given by ( ϕ i , Γ), with i = 1 , . . . , L representing the individual states and Γ representing the existence of a connection betweenmaster node and node i .We have two different types of interaction. First of all we identify the interaction betweenparticle i and its neighbors i − i + 1 with the state of particle i dependent on thestate of its neighbors ( ϕ i − , ϕ i +1 ). In the absence of a master, particle i will align with themajority in the neighborhood, a situation which lead to consensus. If there are differencesbetween neighbor states (frustration), the decision is probabilistic. If the particle i is in thestate s , she/he switches to another state with probability w s . When there is a mixed state(0 | i .The second type occurs with presence of the master node Γ i = 1, which belief or orienta-tion is equal to 1. The probability of the particles being influenced by the master particle isequal to r . Suppose the particle i is in the state s = 1. If most of your neighbors are in thestate s = 0 then the particle will change to state s = 0 with probability 1 − r . However, ifthere is frustration between the neighbors (0 |
1) or (1 | i changes to thestate ( s = 0) with probability r . Now suppose that the particle i is in the state s = 0. Inthe case where neighbors are in the state s = 1, due to peer pressure (majority) and also dueto the influence of the master, the particle will change to state s = 1. There are two conflictsituation. When the majority of neighbors are in the state s = 0, with probability q the par-ticle i will change to state s = 1 or remain in the same state with probability p = 1 − q . Thesecond situation there is frustration between neighbors. Now, with probability r particle i will change to state s = 1 and stay in the same state with probability r = 1 − r . FIG. 1: Model representation. Small spheres represent interacting particles or agents. Eachparticle is in the state s = 0 , w s .Large sphere respresent the master state, inwhich the interaction strength with particles is fixed (denoted by r ) and the links are quencheddisorder with density denoted by ρ . II. THE SECOND NOTATION
In this point we shall change the notations to contruct the new stochastic operator in themore clean way. We write now the global state of the system by (cid:126) s = (s ; s ; . . . ; s L ) , (1)where s i = ( c i , f i ). The variable c i assume the value when the citizen at the site i is inthe moral state, or the value when the citizen is in the immoral state. Besides that, if atsite i there is a fiscal (Regulations, Laws, Norms, Contracts, etc) we write f i = 1 (YES),otherwise f i = 0 (NO).The dynamic at site i is not directly influenced by the presence of fiscals in the neighbor-hood i − i + 1. At each time step (∆ t = 1 /L ) a site i , subject to fiscalization f i = (cid:15) , ischosen randomly among L sites of the lattice. The transition probabilitity of citizen in thestate c i = c out goes to state c i = c in at the time t + ∆ t , given that the states of neighborhood c i − = c L and c i +1 = c R are remained unchanged and the state of fiscals P ( c i = c in ; t + ∆ t | c i − = c L , c i = c out , c i +1 = c R , f i = (cid:15) ; t ) := P c L c R c out c in ( (cid:15) ) . (2)In Table 1 we show the transition probabilities for situations where an individual locatedin a certain i site is not influenced by supervision in the evolution of his state. TABLE I: Probability Transition without fiscalization c out → c in c L c R → → → →
10 0 1 0 1 01 1 0 1 0 10 1 1 − w w w − w In terms of the notation (2) we have P c L c R c out c in (0) = (1 − c in )(1 − c L )(1 − c R ) + c in c L c R ++ (cid:104) (1 − w )(1 − c out )(1 − c in ) + w (1 − c in ) c out ++ w (1 − c out ) c in + (1 − w ) c out c in (cid:105) | c L − c R | . (3)While in table 2 we show the transition probabilities for situations where an individuallocated in a certain i site is influenced by supervision in the evolution of his state. And inthe same way, we have TABLE II: Probability Transition with fiscalization c out → c in c L c R → → → →
10 0 1 − q q − r r − r r r − r P c L c R c out c in (1) = (cid:104) (1 − q )(1 − c out )(1 − c in ) + (1 − r ) c out (1 − c in ) ++ q (1 − c out ) c in + rc out c in (cid:105) (1 − c L )(1 − c R ) + c in c L c R ++ (cid:104) (1 − r )(1 − c out )(1 − c in ) r (1 − c in ) c out + r (1 − c out ) c in ++(1 − r ) c out c in (cid:105) | c L − c R | . (4) Parameterization
Let us choose some constraints to the parameters p, q, r , r , w and w in terms of r (”influence of master node”) and ∆ = w − w , which is the intrinsic state tendency of agents,and ρ . For simplicity, we take q = r . Since w = 1 − w we may write∆ = w − w = 1 − w . (5)The parameter ∆ measures the natural nature of an element or particle be in the state s = 0(1) when ∆ > <
0) in the absence of any interaction or influence. If 0 < w < the individuals, in average, will behave against the norm or the common good. In this case,∆ > < ∆ < r and r should be parameterized so that when the master’s influenceis null ( r = 0) we have r = w and r = w . Otherwise, when r = 1 we should havenecessarily r = 1 and r = 0. The simplest way is through a linear parameterization r = r + (1 − r )( 1 − ∆2 ) , (6) r = (1 − r )( 1 + ∆2 ) . (7)The parametrized version of the model has only three free parameters:0 ≤ r ≤ , ≤ ∆ ≤ ≤ ρ ≤ . (8)Taking into account such parametrizations, the (3) e (4) equations can be grouped in a morecompact form, that is, P c L c R c in ( f i ) = [(1 − rf i )(1 − c in ) + rf i c in ](1 − c R )(1 − c L ) + c in c L c R ++ (1 − rf i )(1 + ∆ − c in ) + 2 rf i | c L − c R | . (9) III. QUANTUM CHAIN
The time evolution of the probability | P ( t ) (cid:105) correspondent to stochastic state | β (cid:105) at time t is governed by markovian transfer operator W [45]. Writing the master equation in itscontinuous-time differential form, we have ∂t | P ( σ, t ) (cid:105) = (cid:88) β w ( β → σ ) | P ( β, t ) (cid:105) − w ( σ → β ) | P ( σ, t ) (cid:105) , (10)where σ, β represent two distinct lattice configuration. Rewriting the equation (10) in itsvector form [44] ∂ t | P (cid:105) = − W | P (cid:105) , (11)where W is a matrix operator, responsible for connecting differents configurations of thevector space. It is also important to mention that, in general, this operator is not Hermitian,i.e., it has complex eigenvalues. These eingenvalues correspond to the oscillations in themodel (imaginary part), while the exponential decay is contained in the real part.In an orthonormal basis we have (cid:104) σ n | | β n (cid:105) = δ σ ,β δ σ ,β · · · δ σ n ,β n . This suggests that we canwrite | P (cid:105) as | P (cid:105) = (cid:88) β P ( β, t ) | β (cid:105) . (12)If we denote the initial probability of the system by | P o (cid:105) = (cid:80) β P o ( β ) | β (cid:105) , the formal solutionof the problem can be written as | P (cid:105) = | P o (cid:105) = e − W t | P o (cid:105) . (13)Due to conservation of probability, we have (cid:104) | W = 0, where (cid:104) | = (cid:80) β (cid:104) β | . Thus anyobservable can be calculated as follows < X > t = (cid:88) β X ( β ) P ( β, t ) | β (cid:105) = (cid:104) | X | P (cid:105) = (cid:10) (cid:12)(cid:12) Xe − W t (cid:12)(cid:12) P o (cid:11) . (14)Here, we always can choose a physical intuitive ”Canonical Base” B = {| β (cid:105) , | β (cid:105) . . . , | β (cid:105) L } to construct the Hilbert Physical Space, i.e; | β (cid:105) = | β (cid:105) (cid:79) | β (cid:105) (cid:79) . . . (cid:79) | β (cid:105) L , (15)where | β (cid:105) i = | β I (cid:105) i ⊗ | β F (cid:105) i . The letter I represents an individual and the letter F representsa fiscal. The vector | β I (cid:105) i can be takes on the number 1 when the individual at the site i is in the moral state and 0 when this is in immoral state. If the site i has a fiscal we willrepresent this sitituation writting | β F (cid:105) i = 1, otherwise | β F (cid:105) i = 0.We now introduce the new stochastic operator related with this model. For instance, wewill design the operator W like (cid:100) W ℵ ρ, ∆ ,r = L − (cid:88) k =2 (cid:100) W ℵ k , (16)where (cid:100) W ℵ k connects two differents states in the assyncronous dynamics, in others words,the matrix elements can be write as (cid:100) W ℵ k = (cid:98) (cid:79) · · · (cid:79) (cid:99) Ω ℵ k (cid:79) · · · (cid:79) (cid:98) . (17)The operators (cid:99) Ω ℵ k act in the state | β (cid:105) k . Assuming periodic boundary conditions ( | β (cid:105) ≡| β (cid:105) L and ( | β (cid:105) L +1 ≡ | β (cid:105) ), we can separate the element (cid:104) α | (cid:100) W ℵ k | β (cid:105) in two contribution,i.e, a contribution (cid:104) α | (cid:100) W ℵ k | β (cid:105) NF without fiscal at the site k and a another contribuiton < α | (cid:100) W ℵ k | β (cid:105) F with a fiscal at the site k . The general matrix element can be write as (cid:104) α | (cid:99) Ω k | β (cid:105) = (cid:104) α | (cid:91) Ψ NFk ⊗ (cid:98) | β (cid:105) + (cid:104) α | (cid:99) Φ Fk | β (cid:105) (18)If we use the Schutz’s prescription [44] to construct the stochastic operator in terms ofPauli’s matrices, then the operator (cid:100) W ℵ k assumes a more elgant form (cid:100) W ℵ k = (cid:88) µ,ν,γ = ± J off[ˆ v ˆ b k ] (cid:101) a µk (cid:103) a νk − (cid:93) a γk +11 + J on[ r ˆ n ˆ b k ] a µk (cid:103) a νk − (cid:93) a γk +11 + a µk a νk − a γk +11 ++[ f (∆) J off[ˆ v ˆ b k ] + g (∆) J off[ˆ v ˆ b k ] a µk + J on[ r ˆ n ˆ b k ] ] (cid:0) a νk − − a γk +1 (cid:1) , (19)where (cid:101) a νk = 1 − a νk , (20) J on[ r ˆ n ˆ b k ] = r ˆ n ˆ b k , (21) J off[ˆ v ˆ b k ] = ˆ v ˆ b k = 1 − r ˆ n ˆ b k , (22) f (∆) = 1 + ∆2 and (23) g (∆) = − ∆ , with [ b + , b − ] k = 0 . (24)0Here the operators a + , a − act in the subspace | β I > and the operators b + , b − act in thesubspace | β F > . However, the most beautiful interpretation is about means of J on and J off. We can roughly look at couplings as a kind of function of the ”quenched impurityinteraction”. In the other words, this simple model revels a new cathegory of interactionsin Statistical Mechanics, i.e; the ON-OFF Quenched Interaction.In addition, if we just ”preserve” the mathematical structure of this operator and choosean appropriate distributions f (∆) and g (∆) with ∆ and r ∈ (cid:60) , then it is possible, for a ± k = σ xk + σ xk , to map the operator (cid:100) W ℵ onto a new class of (cid:91) H XY models, i.e; the (cid:91) H ℵ XY Queched Model with Special Topological Interactions.
IV. CONCLUSION
In the present work we proposed a stochastic quenched disorder model to investigate thepower of a master node over a system formed by L elements disposed in a ring networkwith first neighbor interaction. Due the map between the Master Equation [46] and theSchr¨odinger equation [45, 47] it is possible connect a stochastic one-dimensional model ina quantum chain model. Through the Schutz’s protocol [44], we got map the stochasticoperator in a new XY quenched model wtith special exotic (on-off) interactions. All thequestions addressed go beyond the parametrization studied here.Although the rules of interaction are simple, we uncover a rich scenario of collectivebehaviors. The major evidence is given by the phase diagram presented in [43]. The modelanalyzed here shows the existence of critical values in several parameters. We try to illustratethe volume of the phase space which the coordinates are the control parameter ∆ , r and ρ . In the inner part of this volume the order parameter reach its maximum value M = 1.The shape in this figure is just illustrative. What calls our attention is the properties of thesurface of this volume: it separates the synchronized phase where every elements enter inthe absorbing state s = 1 and the phase where there is a mixture 0 < M <
1. This ideais corroborated by Figure [43]. We fixed a plan by choosing specific values of ∆. After, wevaried ρ and r and we found a critical line splitting two phases. This imply the existence ofa critical surface in the 3D phase diagram.The critical exponents Λ c along the manifold surface likely are non universals since theymay exhibit a continuous dependence of the exponents with the critical control parameters1Λ c (∆ , r, ρ ). This phenomena is represented by small lines leaving the critical surface [43]just to give some ideal of a richness of the phase transition occurring in this system. FIG. 2: Illustrative view of a critical surface. The lines leaving the critical surface illustratedependence of the critical exponent with parameters ∆, 1 − ρ and r . V. ACKNOWLEDGMENTS
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