$(H,ρ)$--induced dynamics and large time behaviors
F. Bagarello R. Di Salvo F. Gargano F. Oliveri






l1,l3,l4]F. Bagarello
fabio.bagarello@unipa.it

l2]R. Di Salvo
rosa.disalvo@unime.it

l1]F. Gargano
francesco.gargano@unipa.it

l2]F. Olivericor
francesco.oliveri@unime.it
[cor]Corresponding author

[l1]DEIM –  University of Palermo,
Viale delle Scienze, I–90128  Palermo, Italy
[l2]Department MIFT – University of Messina,
Viale F. Stagno d'Alcontres 31, I–98166 Messina, Italy
[l3]I.N.F.N -  Sezione di Napoli
[l4]Department of Mathematics and Applied Mathematics,
University of Cape Town, South Africa



In some recent papers, the so called  (H,ρ)-induced dynamics of a system 𝒮 whose time evolution is deduced adopting an operatorial approach, borrowed in part from quantum mechanics, has been introduced. Here, H is the Hamiltonian for 𝒮, while ρ is a certain rule applied periodically (or not) on 𝒮. The analysis carried on throughout this paper  shows that, replacing the Heisenberg
dynamics with the (H,ρ)-induced one, we obtain a simple, and somehow natural, way to prove that some relevant dynamical variables of 𝒮 may converge, for large t, to certain asymptotic values. This can not be so, for finite dimensional systems, if no rule is considered. In this case, in fact, any Heisenberg dynamics implemented by a suitable hermitian operator H can only give an oscillating behavior. We prove our claims both analytically and numerically for a simple system with two degrees of freedom, and then we apply our general scheme to a model describing a biological system of bacteria living in a two-dimensional lattice, where two different choices of the rule
are considered.


Operatorial models Schrödinger and Heisenberg dynamics (H,ρ)–induced dynamics Stressed bacterial populations.

[2010] 37M05 37N20 47L90.








§ INTRODUCTION



Since 2006 <cit.>, it has become clear that raising and lowering operators of quantum mechanics can be  successfully used for the mathematical description of some macroscopic systems.
Many applications are discussed in <cit.>, and others, not necessarily related to ladder operators but still to quantum ideas, have been recently considered
in order to model the dynamics in several contexts, such as social life and decision-making processes <cit.>, population and crowd dynamics <cit.>, ecological processes <cit.>,  political systems <cit.>.

The key aspect of the operatorial approach adopted here is that the time evolution of an observable X of the macroscopic system 𝒮 we are considering is given by X(t)=exp(iHt)Xexp(-iHt), where H is the time independent self-adjoint Hamiltonian of 𝒮, and  the mean values of the observable X are linked to real valued functions phenomenologically associated to some macroscopic quantities. This approach has revealed efficient in predict real dynamics.

Not surprisingly, the description of the dynamics of operatorial models ruled by a self-adjoint time independent Hamiltonian has some limitations. For instance, all the observables of a system 𝒮 with a finite number of degrees of freedom and governed by a quadratic Hamiltonian operator, exhibit, at most, a quasiperiodic dynamics. Then, if we wish to model a system  𝒮 having an asymptotic final state, it is clear that such a description does not work, and the modeling framework needs to be enriched, if not completely changed.
This is not new: in quantum optics, or for two or three level atoms, if we need to describe a transition from one level to another, in some cases  some effective finite dimensional non-hermitian Hamiltonians are used. In this way, decays are well described phenomenologically. Another way consists in considering the atoms interacting with some infinite reservoir, but  in such a case the full system
(𝒮_full, i.e., 𝒮 plus the reservoir) is no longer finite-dimensional. Sometimes one describes a simplified version of 𝒮_full by means of some master equation where, again, the dynamics is described by some finite matrix whose entries are properly chosen; per contra,  many details of the system are lost with this kind of description, and the physics could be somehow hidden.

In  <cit.>, where a quantum version of game of life has been considered, we have proposed an extended version of the Heisenberg dynamics in order to take into account effects which may occur during the time evolution of the system 𝒮, and which can not apparently be included in a purely Hamiltonian description. In particular, at fixed times of the evolution of a system 𝒮 driven by a hermitian  time independent Hamiltonian H, some checks on 𝒮 are performed, and  used to change the state of the system itself according to some explicit prescription.  
A slightly different viewpoint has been proposed in <cit.>, where the periodical checks on the state of the system are used to change some of the parameters entering the Hamiltonian, without modifying the functional form of the Hamiltonian itself. This approach proved to be quite efficient in operatorial models of stressed bacterial populations <cit.>, as well as in models of political parties affected by turncoat–like behaviors <cit.>.

Here, we want to show that this (H,ρ)–induced dynamics can be efficiently used to describe (finite-dimensional) systems going to some equilibrium, even when H is hermitian. This strategy may give interesting results if the rule ρ is not introduced as a mere mathematical trick, but  is somehow physically justified. For instance,  in <cit.>, the rule accounts for the modifications in the metabolic activity of bacteria due to lack of nutrients and/or to the presence of waste material, whereas, in <cit.>, the rule modifies the behavior of the members of a political party with regard to their tendency to shift allegiance from one loyalty or ideal to another one.

The paper is organized as follows. In Section <ref>, we briefly review the main definitions and results concerning the two possible strategies for  (H,ρ)–induced dynamics. In Section <ref>, we describe a very simple two-mode system, and discuss the role of the rule ρ in the emergence of an asymptotic time evolution.  Section <ref> is devoted to the analysis of a more complicated system, i.e., a model for a bacterial population on a two-dimensional lattice.
Section <ref> contains our conclusions and perspectives. Finally, in the appendix, we consider a situation where 
a simple time dependent Hamiltonian may produce a dynamics with an asymptotic state, at least under very special choices of its parameters.




§ EXTENDING THE DYNAMICS WITH RULES: (H,Ρ)-INDUCED DYNAMICS



To keep the paper self-contained, in this Section we briefly discuss the mathematical framework of the (H,ρ)–induced dynamics (further details can be found in <cit.>). 
The proposed approaches merge the general framework of quantum dynamics, described by a Hamiltonian H, with some periodic (or not) effects which can not be included in H.

Let 𝒮 be our physical system, and Q_j (j=1,…,M) a set of M commuting self-adjoint operators with eigenvectors φ^(j)_α_n and eigenvalues α_n^(j):

    [Q_j,Q_k]=Q_jQ_k-Q_kQ_j=0,      Q_j=Q_j^†,      Q_jφ^(j)_n_j=α_n_j^(j)φ^(j)_n_j,

j,k=1,2,…,M, n_j=1,2,3,…,N_j. Setting 𝐧=(n_1,n_2,…,n_M), the vector

    φ_𝐧=φ^(1)_n_1⊗φ^(2)_n_2⊗⋯φ^(M)_n_M

represents an eigenstate of all the operators Q_j, say

    Q_j φ_𝐧=α_n_j^(j) φ_𝐧.

The existence of a common eigenstate for all the operators Q_j is guaranteed by the fact that they all mutually commute. It is convenient, and always true in our applications, to assume that these vectors are mutually orthogonal and normalized:

    <φ_𝐧,φ_𝐦>=δ_𝐧,𝐦=∏_j=1^Mδ_n_j,m_j.

The Hilbert space ℋ where 𝒮 is  defined is mathematically constructed as the closure of the linear span of all the vectors φ_𝐧,  defining an orthonormal basis for ℋ. Now, let H be a quadratic
time independent self-adjoint Hamiltonian, describing the interactions occurring in 𝒮, plus its free kinetic contribution. Notice that H, in general, does not commute with the Q_j's. In absence of any other information, the wave function
Ψ(t), describing 𝒮 at time t, evolves according to the
Schrödinger equation iΨ̇(t)=HΨ(t), where Ψ(0)=Ψ_0 describes the initial state of 𝒮.
The formal solution of the Schrödinger equation is, since H does not depend explicitly on t, Ψ(t)=exp(-iHt)Ψ(0)=exp(-iHt)Ψ_0.
We can now compute the mean value of each operator Q_j in the state Ψ(t),

    q_j(t)=<Ψ(t),Q_jΨ(t)>,

and use it to define a related M-dimensional time dependent vector 𝐪(t)=(q_1(t), q_2(t),…, q_M(t)).

It is well known <cit.> that this is not the unique way to look at the time evolution of 𝒮. Another equivalent way consists in adopting the Heisenberg representation, in which the wave function does not evolve in time, while the operators do, according to the equation Ẋ(t)=i[H,X(t)], whereupon we have X(t)=exp(iHt)X(0)exp(-iHt). Here, X(t) is a generic operator of 𝒮 acting on ℋ at time t. In this approach, if X is an observable of the system, after fixing a vector state
φ_𝐧 representing the initial configuration of the system, we may compute its mean value

    x(t)=<φ_𝐧,X(t)φ_𝐧>,

that, in the case of fermionic operatorial models, if X is the number operator, can be interpreted as a  local density <cit.>.

We are now ready to introduce, rather generally, two different ways of extending the dynamics through the introduction of a specific rule.

In the first approach, the rule ρ is a  map from ℋ to ℋ.
Its explicit action depends on the expression of 𝐪(t) at particular instants kτ (k∈ℕ, τ∈ℝ^+). In other words, according to how 𝐪(kτ) looks like, ρ maps the input vector Ψ(τ) (here k=1) into a different output vector Ψ_new, and we write  ρ(Ψ(τ))=Ψ_new.
This is not very different from what happens in scattering theory, where an incoming state, after the occurrence of the scattering, is transformed into an outgoing state <cit.>.  The new vector Ψ_new can be considered as the new initial state of the system driven, for another time interval of length τ, by the Hamiltonian H, and this procedure can be iterated several times. As an example, in <cit.>, the rule ρ is used to map an incoming state into one of the elements of the basis of ℋ, allowing to extend the classical concept of rule used in a cellular automaton. Our approach has some aspects in common with the repeated quantum measurements, according to which the state of a system is perturbed by a non trivial quantum measurement, and subsequent measurements of the same system reveal the presence of disturbances if compared to the situation in which no previous measurements were performed. In our case, the disturbance manifests with the creation of the new state  Ψ_new. The key difference in our approach lies in the fact that
the new generated state is prepared as the result of the choice of the rules, so that it is the result of a controlled post-processing phase. On the other hand, in the classical notion of repeated quantum measurements, the new state is the result of disturbances that may not be controlled depending on the kind of apparatus used for the measurements <cit.>. Here, we also like to mention that the repeated action of the rule ρ in our approach is similar to what is done in the context of the quantum Zeno effect (see <cit.>, and references therein), where some measures on a quantum system are repeated again and again.

In the second approach, the rule ρ works on the space of the parameters of the Hamiltonian H, rather than on the wave function. In this case, given a quadratic time independent
self-adjoint Hamiltonian involving p real parameters, the rule ρ is a map 
from ℝ^p to ℝ^p
that, at particular instants kτ, on the basis of the actual state of the system (or of its variation), changes
some of the values of these parameters. In such a way, in some sense, the model adjusts itself.



 §.§ The rule ρ as a map from ℋ to ℋ


Let us briefly sketch some general considerations about the rule ρ  as a map from ℋ to ℋ.
First, let us observe that there exists a one-to-one correspondence between 𝐧 and the vector φ_𝐧:
once we know 𝐧, φ_𝐧 is clearly identified, and  vice-versa. Suppose now that at
time t=0 the system 𝒮 is in a state 𝐧^0 or, which is the same, 𝒮 is described by the vector
φ_𝐧^0. Then, once fixed a positive value of τ, this vector evolves
 in the time interval [0,τ[ according to the Schrödinger recipe: exp(-iHt)φ_𝐧^0. Let us set

    Ψ(τ^-)=lim_t→τ^-exp(-iHt)φ_𝐧^0,

where t converges to τ from below[We use here τ^-, 2τ^-, …,  as argument of Ψ to emphasize that before τ, for instance, the time evolution is only due to H, while ρ really acts at t=τ.]. Now, at time t=τ, ρ is applied to Ψ(τ^-), and the output of this action is a
new vector which we assume here to be again an eigenstate of each operator Q_j, but with different eigenvalues, φ_𝐧^1[This choice is not the only possibility to set up a rule. In fact, other possibilities can also be considered. The key common point to all possible choices is that ρ behaves as a check over the system 𝒮, and modifies some of its ingredients according to the result of this check. See also Section <ref>.]. In other words, ρ looks at the  explicit expression of  Ψ(τ^-) and, according to its form, returns a new vector 𝐧^1=(n^1_1,n^1_2,…,n^1_M);  as a consequence, a
new vector φ_𝐧^1 of ℋ is obtained. Examples of how ρ explicitly acts are discussed in Section <ref>. Now, the procedure is iterated, taking φ_𝐧^1
as the initial vector, and letting it evolve with H for another time interval of length τ;
we compute

    Ψ(2τ^-)=lim_t→τ^-exp(-iHt)φ_𝐧^1,

and the new vector  φ_𝐧^2 is deduced by the action of rule ρ on Ψ(2τ^-): φ_𝐧^2=ρ(Ψ(2τ^-)).   Then, in general, for all k≥1, we have

    Ψ(kτ^-)=lim_t→τ^-exp(-iHt)φ_𝐧^k-1,    φ_𝐧^k=ρ(Ψ(kτ^-)).


Now, let X be a generic operator on ℋ, bounded or unbounded. In this latter case, we will require that the various φ_𝐧^k  belong to the domain of X(t)=exp(iHt)Xexp(-iHt) for all t∈[0,τ].


The sequence of functions

    x_k+1(t):=<φ_𝐧^k, X(t)φ_𝐧^k>,

for t∈[0,τ] and k∈ℕ_0, is called the (H,ρ)–induced dynamics of X.


Some consequences of Definition <ref> and some properties of the sequence X(τ)=(x_1(τ),x_2(τ),…) have been discussed in <cit.>. Moreover,
 from X(t)=(x_1(t), x_2(t),…) it is possible to define a new function of time, giving the evolution of the observable X of 𝒮, in the following way:

    X̃(t)=
    {[       x_1(t)  t∈ [0,τ[  ,;     x_2(t-τ) t∈ [τ,2τ[  ,;    x_3(t-2τ) t∈ [2τ,3τ[ ,;            …              ].

It is clear that X̃(t) may have discontinuities in kτ, for positive
integers k.




 §.§ The rule ρ as a map in the space of the parameters of H


In the previous subsection, the effect of ρ was to change the state of the system, from an input to an output vector. The other elements of 𝒮, in particular its Hamiltonian, stay unchanged at each step. We now discuss a different approach where a rule acts on 𝒮 changing some aspects of the dynamics of 𝒮 related to the Hamiltonian operator.  

Let 𝒮 be a system involving M fermionic (or bosonic) modes and suppose that its evolution is ruled by the following quadratic time independent self-adjoint Hamiltonian

    H=∑_j=1^Mω_j a_j^† a_j+∑_j=1^M-1∑_k=j+1^Mλ_j,k(a_ja_k^†+a_ka_j^†),

involving the p=M(M+1)/2  real parameters (not necessarily all non vanishing) ω_j and λ_j,k, where a_j and a_j^†, i=1,… M, are annihilation and creation operators, respectively.

Adopting the Heisenberg representation, the time evolution of the lowering operators a_j's is given by

    a_j(t)=exp(iHt)a_j(0)exp(-iHt),     j=1,…,M,

or,
equivalently, by the solution of the following linear system of ordinary differential equations:

    ȧ_j(t)=i(-ω_j a_j(t)+∑_k=1,k≠ j^M λ_j,ka_k(t)),
         j=1,…,M.


Restricting ourselves to the fermionic case, in principle, we have a system of M 2^2M linear differential equations to be solved[For bosons the situation is, in general, more tricky since we have to deal with unbounded operators in an infinite-dimensional Hilbert space.] with suitable initial conditions for the matrices representing the operators a_j. However, since the system is linear, we may write it in compact form, say

    Ȧ(t)=U A(t),

where A(t)=(a_1(t),  a_2(t), …, a_M(t))^T,
and U is an M× M constant matrix such that U_j,j=-iω_j,
U_j,k=iλ_j,k, and each component of A is a 2^M× 2^M  matrix. The formal solution is immediately deduced, namely

    A(t)=exp(U t)A(0)=V(t)A(0).

Thus, if V_ℓ,m(t) is the generic entry of matrix V(t), we have

    a_ℓ(t)=∑_k=1^M V_ℓ,k(t)a_k(0),     ℓ=1,…,M.


Now, we need to compute the mean value of the number operator for the ℓ-th mode (which is intended to represent a physical quantity which is relevant for the description of 𝒮)

    n̂_ℓ(t)=a_ℓ^†(t)a_ℓ(t)

on an eigenvector φ_n_1,n_2,…,n_M of all the n̂_ℓ(0),

    n̂_ℓφ_n_1,n_2,…,n_M=n_ℓφ_n_1,n_2,…,n_M,     ℓ=1,2,…,M.

It is easy to check that the relations

    n_ℓ(t)=<φ_n_1,n_2,…,n_M,n̂_ℓ(t)φ_n_1,n_2,…,n_M>=
    ∑_k=1^M |V_ℓ,k(t)|^2n_k,

where ℓ=1,…,M, provide what we are looking for.

Starting from a quadratic Hamiltonian such as the one defined in (<ref>), we may enrich the dynamics by introducing a rule, repeatedly acting at specific instants, and accounting for a sort of dependence of the parameters ω_j and λ_j,k in (<ref>) upon the current state of the system.   In some sense, the model adjusts itself as a consequence of its evolution. We stress that in such an approach only the strengths of the mutual interactions change, whereas the model preserves its functional structure. 

Here is a sketch of this approach. Let us start considering a self-adjoint quadratic Hamiltonian operator H^(1), the corresponding evolution of a certain observable X

    X(t)=exp(iH^(1)t)Xexp(-iH^(1)t),

and compute its mean value

    x(t)=⟨φ_n_1,n_2,…,n_M, 
    X(t)φ_n_1,n_2,…,n_M⟩

in a time interval of length τ>0 on a vector φ_n_1,n_2,…,n_M. Then, let us modify some of the parameters involved in H^(1), on the basis of the values of the various x(τ) according to (<ref>). In this way, we get a new Hamiltonian operator H^(2), having the same functional form as H^(1), but (in general) with  different values of (some of) the involved parameters, and follow the continuous evolution
of the system[The evolution of 𝒮 is no more stopped and restarted from a modified state, as done in <ref>, but rather the initial vector state for each subinterval corresponds exactly to the state reached by the system.] under the action of this new Hamiltonian for the next time interval of length τ. 
Actually, we do not restart the evolution of the system from a new initial condition, but simply continue to follow the evolution with the only difference that for t∈]τ,2τ] a new Hamiltonian H^(2) rules the process. 
And so on. Therefore, the rule now has to be thought of as a map from ℝ^p into ℝ^p acting on the space of the parameters involved in the Hamiltonian. Therefore, the global evolution is governed by a sequence of similar Hamiltonian operators, and the parameters entering the model can be considered stepwise (in time) constant.

In general terms, let us consider a time interval [0,T], and split it in n=T/τ subintervals of length τ. Assume  n to be integer. In the k-th subinterval [(k-1)τ,kτ[ consider an Hermitian Hamiltonian H^(k) ruling the dynamics. The global dynamics arises from the sequence of Hamiltonians

    H^(1)τ⟶ H^(2)τ⟶ H^(3)τ⟶…τ⟶ H^(n),

the complete evolution being obtained by glueing the local evolutions.

In every subinterval we therefore have a system like

    Ȧ(t)=U^(k)A(t),       t ∈ [(k-1)τ,kτ].


To obtain the mean values of the number operators at each instant t, we need the computation of the exponential of the M× M matrix U^(k) t.
There are various methods in the literature, based on different grounds (see <cit.>, and the references therein, for a survey on this topic), for carrying out the exponential of large matrices.
The best general algorithms use matrix decomposition methods;
they start with the Schur decomposition and include some sort of eigenvalue
clustering. There are also variants which involve further reduction to a block form. In all
cases the initial decomposition costs O(M^3). Therefore, roughly speaking, the computational complexity
for the exponential of a matrix of order  M can be considered as O(M^3).

System (<ref>) is made of M first order linear differential equations, and a good
numerical solution (for instance by a Runge-Kutta method) at the cost O(M^2) (O(M) if U^(k) is a sparse matrix) could be looked for. Nevertheless, this is not completely true since a numerical approach requires the use of A(0), and each component of  the latter is a matrix  of order 2^M; therefore, the linear system (<ref>) actually involves M2^2M differential equations.

We now discuss a different point of view for the same problem that bypasses both the need of directly computing the exponential of the matrix U^(k) t, and the huge amount of computation to obtain a numerical solution of (<ref>), according to the classical representation of M-mode fermionic operators. 

Let us reconsider the system (<ref>) but assuming that  each component A_j(t) of A is an M-component row vector whose value at t=0 is
the j-th element of the canonical orthonormal basis of ℝ^M.
As a consequence,  (<ref>) now represents a linear differential equation for the M× M matrix A to be solved with the initial condition A(0)=, and now the solution reads

    A(t)=exp(U^(k)t);

we can compute this solution by numerically solving a system of M^2 linear differential equations, with computational cost O(M^3) that reduces to O(M^2) if the matrix U^(k) is sparse (this usually occurs in
operatorial models on a lattice <cit.>).

The expression |A_jk(t)|^2 provides the mean value n_j(t) corresponding to the initial values n_ℓ(0)=δ_kℓ (ℓ=1,…,M).
Therefore, in correspondence to the general initial values (n_1,…,n_M), the mean values are obtained
by means of the formula

    n_ℓ(t)=|A_ℓ,1(t)|^2n_1+|A_ℓ,2(t)|^2n_2+⋯|A_ℓ,N(t)|^2n_M.


We point out that this approach in the case where U^(k) is not a sparse matrix does not  lead
to a relevant lowering of computation complexity. Nevertheless, this strategy becomes essential if the evolution of the system is governed by a Hamiltonian operator together with the periodic application of a rule modifying the values of some of the parameters involved in the model on the basis of the current state of the system. In fact, 
with such an approach, we have

    A(t)=exp(U^(k)t),     t ∈ [(k-1)τ,kτ].

Then, glueing the solutions in all the subintervals, we find

    A(t)={[                            exp(U^(1)t)                                                                    t ∈ [0,τ],;             exp(U^(2)(t-τ))exp(U^(1)τ)                                                                   t ∈ [τ,2τ],; exp(U^(3)(t-2τ))exp(U^(2)τ)exp(U^(1)τ)                                                                  t ∈ [2τ,3τ],;                                      …                                                                             … ].


This kind of rule-induced stepwise dynamics clearly may generate discontinuities in the first order derivatives of the operators, but prevents the occurrence of jumps in their evolutions and, consequently, in the mean values of the number operators.
By adopting this rule, we are implicitly considering the possibility of having a time dependent Hamiltonian. However, the time dependence is, in our case, of a very special form: in each interval [(k-1)τ,kτ[ the Hamiltonian does not depend on time, but in kτ some changes may occur, according to how the system is evolving. For this reason, our Hamiltonian can be considered piecewise constant in time. A comparison of this approach with that related to an explicitly time dependent Hamiltonian is discussed in Appendix 1.




§ A TWO-MODE SYSTEM



In this Section, by considering a very simple toy model introduced in <cit.>, we will discuss the possibility of getting, in a simple and natural way, a dynamics approaching an asymptotic equilibrium state.

Let us consider a system 𝒮, having two (fermionic) degrees of freedom, and the Hamiltonian

    H=H_0+λ H_I,     H_0=ω_1a_1^† a_1+ω_2a_2^† a_2,    H_I=a_1^† a_2+a_2^† a_1,

where ω_j and λ are real (and positive) quantities in order to ensure that H is self-adjoint. The operators a_j and a_j^† are assumed to satisfy the following anticommutation rules:

    {a_i,a_j^†}=δ_i,j ,    {a_i,a_j}={a_i^†,a_j^†}=0,

i,j=1,2, where, as usual,  is the identity operator and {x,y}:=xy+yx.
Of course, when λ=0, there is no contribution in H due to mutual interaction.


The eigenstates of the number operators n̂_j:=a_j^† a_j are easily obtained: if φ_0,0 is the ground vector of 𝒮,
a_1φ_0,0=a_2φ_0,0=0, an orthonormal basis of our four-dimensional Hilbert space ℋ is given by

    φ_0,0,    φ_1,0:=a_1^†φ_0,0,     φ_0,1:=a_2^†φ_0,0,     φ_1,1:=a_1^† a_2^†φ_0,0.

We have 

    n̂_1φ_n_1,n_2=n_1φ_n_1,n_2,    n̂_2φ_n_1,n_2=n_2φ_n_1,n_2.


The equations of motion for the annihilation operators a_j(t) are

    ȧ_1(t)=-iω_1 a_1(t)-iλ a_2(t),    ȧ_2(t)=-iω_2 a_2(t)-iλ a_1(t),

that can be solved with the initial conditions a_1(0)=a_1 and a_2(0)=a_2. The solution looks like

    a_1(t)=1/2δ(a_1((ω_1-ω_2)Φ_-(t)+δΦ_+(t))+2λ a_2Φ_-(t)),
       a_2(t)=1/2δ(a_2(-(ω_1-ω_2)Φ_-(t)+δΦ_+(t))+2λ a_1Φ_-(t)),

where

    δ=√((ω_1-ω_2)^2+4λ^2),
       Φ_+(t)=2exp(-it(ω_1+ω_2)/2)cos(δ t/2),
       Φ_-(t)=-2iexp(-it(ω_1+ω_2)/2)sin(δ t/2).

Then, the functions n_j(t):=<φ_n_1,n_2,
n̂_j(t)φ_n_1,n_2> are

    n_1(t)=n_1(ω_1-ω_2)^2/δ^2+ 4λ^2/δ^2(n_1cos^2(δ t/2)+n_2sin^2(δ t/2)), 
       n_2(t)=n_2(ω_1-ω_2)^2/δ^2+ 4λ^2/δ^2(n_2cos^2(δ t/2)+n_1sin^2(δ t/2)).

These functions can be interpreted, in agreement with other applications considered along the years, as the densities of two species, 𝒮_1 and 𝒮_2, interacting as in (<ref>) in a given (small) region. The interaction Hamiltonian H_I in (<ref>) describes a sort of predator-prey mechanism, and this reflects in the solution (<ref>), showing how the two densities, because of the interaction between  𝒮_1 and 𝒮_2, oscillate in the interval [0,1]. Otherwise, if λ=0, n_j(t)=n_j, the densities stay constant, and nothing interesting happens in 𝒮. We observe that the formulae in (<ref>)
automatically imply that n_1(t)+n_2(t)=n_1+n_2, independently of t and λ: the oscillations are such that they sum up to zero. We refer to <cit.> for more details on this model, and for its role in modeling  migration, which is achieved considering a 2D version of the H in (<ref>), with an additional term accounting for the diffusion of the two species in a lattice. Here, we exploit the possibility of getting some limiting values for n_1(t) and n_2(t) for large values of t, when λ≠0.

The first trivial remark is that the functions n_1(t) and n_2(t) in (<ref>) do not admit any asymptotic limit, except when n_1=n_2 
(or when λ=0, which is excluded here). In this case, clearly, n_1(t)=n_2(t)=n_1=n_2. On the other hand, if n_1≠ n_2, then both n_1(t) and n_2(t) always oscillate in time. This is not surprising since it is easy to prove that
if 𝒮 is a system living in a finite dimensional Hilbert space, and if its dynamics is driven by a time independent, self-adjoint, quadratic Hamiltonian H̃, then its evolution is necessarily periodic or quasi-periodic. In fact,
 if H̃=H̃^† is an M× M matrix, then an unitary matrix U exists such that UH̃U^-1=H_d, which is a diagonal matrix with M (not necessarily different) real eigenvalues E_1, E_2, …, E_N. Then, if n̂ is an observable of 𝒮, its time evolution can be written as

    n̂(t)   =exp(iH̃t)n̂exp(-iH̃t)=
       =U^-1exp(iH_dt)(Un̂ U^-1) exp(-iH_dt)U,

which is periodic when all couples of eigenvalues of H̃ are commensurable, while is quasiperiodic otherwise. If H_d commutes with Un̂ U^-1, then n̂(t)=n̂, and its large time behavior is clearly trivial. Otherwise, n̂(t) keeps on oscillating, and no asymptotic value is reached. This is essentially what is described by formulae (<ref>), where the focus is not really on the operators n̂_1 and n̂_2, but on their mean values.



 §.§ The rule as a map from ℋ to ℋ and the existence of an asymptotic value


As we have discussed previously, what is interesting for us is to describe the existence of some asymptotic limit for our system 𝒮. If 𝒮 lives in a finite dimensional Hilbert space, our previous simple argument shows that no time independent, self-adjoint, Hamiltonian H̃ can make the job. Hence, we have to add some extra ingredients, or change something in the Hamiltonian. In fact, it is well known that if 𝒮 interacts with a reservoir with an infinite number of degrees of freedom, <cit.>, such a limit can indeed be obtained. And, in fact, this possibility has been used in several applications also because it admits interesting interpretations, <cit.>. However, from a technical point of view, this is probably not the easiest choice and, in fact, quite often one makes use of some finite dimensional effective Hamiltonian H_eff, with H_eff≠ H_eff^†. For instance, this is what is done in optics <cit.>. Here, we will show how the use of a rule, other than having a concrete meaning, can produce a realistic asymptotic value for the observables of 𝒮.

In order to show this, we first rewrite (<ref>) as

    N(t)=T_tN(0),

where

    N(t)=(
    [ n_1(t); n_2(t);        ]),   
    T_t=1/δ^2(
    [ δ^2-4λ^2sin^2(δ t/2)     4λ^2sin^2(δ t/2);     4λ^2sin^2(δ t/2) δ^2-4λ^2sin^2(δ t/2) ]).

Of course, the components of N(t) return the expressions of n_1(t) and n_2(t) for all times.  Let us now see what happens if we insert a certain rule ρ in the time evolution of the system.

Here, we can think of ρ as a measure of n_1(t) and n_2(t) repeated at time τ, 2τ, 3τ, … We know that performing a measure on a quantum system is a delicate operation, which modifies the system itself <cit.>. Therefore, there is no reason a priori to say that the result of a measure at time kτ (after having measured the system at time τ, 2τ, …, (k-1)τ) would be exactly the same as the one we deduce directly from (<ref>), i.e., n_1(kτ) and n_2(kτ), and in fact this is exactly what we are going to show now.

The first measure gives N_1(τ):=N(τ)=T_τ N(0).  Then, according to what is discussed in Section <ref>, we let the system evolve out of this new initial condition N_1(τ) for another time step: N_2(τ):=T_τ N_1(τ)=T_τ^2 N(0), and so on. It is quite natural to call the rule ρ considered here a stop and go rule: apparently, in fact, ρ just stops the time evolution at τ, 2τ, 3τ and so on, and then let the time evolution start again. Of course, this can be related to some action on the state of the system. Also, it is not hard to imagine richer versions of the rule, where the new state of the system is fixed by some external action, as in Section <ref>.  So doing we produce a sequence

    N_ℓ(τ)=T_τ^ℓ N(0),

for all ℓ≥1.

In order to compute N_ℓ(τ), and its limit for ℓ diverging, we first observe that T_t is a self-adjoint matrix, so it can be easily diagonalized. In particular, we get

    U^-1T_tU=(
    [ λ_1(t)      0;      0 λ_2(t);        ])=:Λ_t,

where

    U=1/√(2)(
    [  1 -1;  1  1;    ]),  λ_1(t)=1,  λ_2(t)=1/δ^2(δ^2-8λ^2sin^2(δ t/2)).

Then

    T_τ^ℓ=UΛ_τ^ℓ U^-1=U(
    [        1        0;        0 λ_2^ℓ(τ);          ])U^-1,

so that N_ℓ(τ)=T_τ^ℓ N(0) can converge if λ_2^ℓ(τ) does converge when ℓ diverges.
This is what happens whenever the parameters δ, τ and λ satisfy the following inequalities:

    0<8λ^2sin^2(δ t/2)<δ^2.

In fact, when this is true, λ_2(τ)∈]0,1[, and, therefore, lim_ℓ→∞λ_2^ℓ(τ)=0. Hence,

    lim_ℓ→∞N_l(τ)=(
    [ n_1(0);      0;        ]),

which clearly shows that a non trivial equilibrium can be reached in this case. However, if the parameters do not satisfy (<ref>), the asymptotic behavior of N_ℓ(τ) can be completely different. In fact, taking, for instance, τ=π/δ and λ=√(3/8) δ, and then fixing δ=1 for simplicity, we deduce that λ_2(τ)=-2, so that lim_ℓ→∞|λ_2(τ)|^ℓ=∞: thus, it is evident that the role of the parameters of H are in fact essential. This is also evident from inequality (<ref>), which is satisfied for all t if λ<(ω_1-ω_2)/2. 

The conclusion of the analysis of this simple example is the following: even in the presence of a self-adjoint Hamiltonian, a simple two-modes fermionic system admits a non trivial asymptotic limit for a large range of values of the parameters of the model, at least if a stop and go rule is assumed. However, the same rule can also produce a non converging dynamics for special choices of the parameters.




 §.§ The rule working on the space of parameters

As we have discussed in Section <ref>, using a rule which modifies the state of the system is not the unique choice. We can also modify the parameters of the Hamiltonian, which is the choice we are going to discuss here. Then,
let us consider again the model with two fermionic modes ruled by the Hamiltonian (<ref>). In this case, let us
consider the (H,ρ̃)–induced dynamics where the rule ρ̃ acts on the space of parameters. More precisely, let us
assume that the rule modifies at fixed times kτ  only the parameters ω_1 and ω_2 in (<ref>) according to the variations of n_1 and n_2 in the interval [0,kτ] (rule ρ̃_A) or in the (smaller) interval [(k-1)τ,kτ] (rule ρ̃_B). Both these rules, see (<ref>) and (<ref>), are clearly completely different from a stop and go rule.




Let us fix the initial values of the parameters, say  ω_1=1/√(3), ω_2=1, λ=1/√(5), and the
initial conditions n_1=0.8, n_2=0.6. Without applying any rule we have (see Figure <ref>), as one expects, a never ending oscillating behavior of both n_1(t) and n_2(t).

By taking the rule

    ρ̃_A(ω_1)=ω_1(1+δ_1),           δ_1=n_1(kτ)-n_1(0),
       ρ̃_A(ω_2)=ω_2(1+δ_2),           δ_2=n_2(kτ)-n_2(0),

we can see in the left part of Fig. <ref> how the system reaches some asymptotic states; also, we see that the rate of decay
of oscillating behaviors is smaller as the value of τ is increased. Notice that, since n_1(t)+n_2(t) is a constant,
δ_1+δ_2=0, so that the variations of the inertia parameters ω_1 and ω_2 are opposite.

If we consider the rule

    ρ̃_B(ω_1)=ω_1(1+δ_1),           δ_1=n_1(kτ)-n_1((k-1)τ),
       ρ̃_B(ω_2)=ω_2(1+δ_2),           δ_2=n_2(kτ)-n_2((k-1)τ),

we can see in the right part of Fig. <ref> how the system tends to reach again some asymptotic states, but in a slightly different way.




The conclusion of the analysis of this simple model is that the rules, both the one defined on ℋ and those working on the space of the parameters, strongly affect the time evolution of the system producing serious consequences. For this reason, we believe they can be a valid alternative to the open system procedure  adopted, for instance, in <cit.>.





§ A MANY-MODE SYSTEM: THE LONG TERM SURVIVAL OF BACTERIAL POPULATIONS IN A SQUARE REGION



In this Section, we consider an operatorial model, which takes some ideas from the one developed in <cit.>, and  investigated in <cit.>,
for the description in a finite 2D region of  bacterial populations belonging to certain genera, such as Bacillus,  Clostridium or Pseudomonas, whose metabolism ensures their long term survival  in terms of latent life when under negative stimuli. In recent years, special attention has been devoted by microbiologists to Pseudomonas aeruginosa, an ubiquitous bacterium capable to use more than one hundred of chemical compounds as carbon and energy sources, and, due to its wide metabolic versatility, to persist for prolonged periods of time without external sources of nutrients
<cit.>. Pseudomonas aeruginosa is the ethiological agent of several diseases, and represents an emergence in hospital-acquired infections, because of its high degree of resistance against several classes of antibiotics and of its persistence also in disinfecting solutions.



To describe colony morphology in stressed/aged bacterial populations, let us consider a region represented by a regular L× L square grid (see Fig. <ref>). In each cell the model is made of four compartments (Fig. <ref>); each compartment is represented by a fermionic operator and, according to the usual interpretation, the mean values of the number operators are used to represent the local densities of the different compartments <cit.>. The first mode is related to the nutrients, the second one to the bacteria, and the last two to a couple of different garbages, see below.
The reason of the introduction of two distinct compartments for the garbage is due to the different roles they play in the ecosystem; in fact, the first one is associated to the dead cells which are rapidly reusable as nutrients, whereas the second one contains the waste material which is not yet recyclable over the short term or no longer reusable at all and, therefore, may act as a a stress factor for the system <cit.>.


Each actor, associated to an annihilation fermionic operator a_j,α (in the following we use latin and greek indices to refer to the compartments, and to the cells, respectively) occupies the cell α and interacts with the other actors in the same cell and in its Moore neighborhood[The Moore neighborhood of a cell in a 2D lattice consists in the eight cells surrounding it.]. The dynamics of the system is governed by the self-adjoint quadratic Hamiltonian operator

    {
    H   =H_0+ H_I+H_M,      with 
    
    H_0    = ∑_α=1^L^2 ∑_j=1^4 ω_j,α  a_j,α^† a_j,α,
    
    H_I    = ∑_α=1^L^2 (
    ∑_j=2^4 λ_j,α(a_1,α a_j,α^†+a_j,α a_1,α^†).
       +.∑_k=3^4 ν_k,α(a_2,α a_k,α^†+a_k,α a_2,α^†)
    ),
    
    H_M    = ∑_α=1^L^2 μ_2,α ∑_β=1^L^2p_α,β (a_2,α a_2,β^†+a_2,β a_2,α^†),
    .

in which j=1,2,3,4 label respectively the nutrients, the bacteria, and the garbages of the two types. We see that H consists, besides the first standard part H_0, of  the two contributions H_I and H_M, which model respectively the interactions among
the different actors, and the migration or diffusion of the bacteria, respectively. All the parameters involved in H are real numbers having the following interpretation: the constants
ω_j,α describe the inertia of all the compartments in each cell (expressing the tendency of each degree of freedom to stay constant in time <cit.>), while λ_j,α (j=2,3,4), ν_k,α (k=3,4) are used to characterize the strength of the interactions among the bacteria, the nutrients, and the garbages;  the remaining parameters p_α,β and μ_2,α are related to the possibility of the bacterial population to move from the cell α to cell β, and to its mobility, respectively.

The linear equations of motion deduced for each operator in accordance with the Heisenberg scheme read

    {ȧ_1,α   =i(-ω_1,α a_1,α+λ_2,α a_2,α+λ_3,α a_3,α+λ_4,α a_4,α),
    ȧ_2,α   =i ( -ω_2,α a_2,α+λ_2,α a_1,α+ν_3,α a_3,α+ν_4,α a_4,α∑_β=1^L^2.
       +.∑_β=1^L^2(μ_2,α+μ_2,β) p_α,β a_2,β),
    ȧ_3,α   =i(-ω_3,αa_3,α+λ_3,α a_1,α+ν_3,α a_2,α),
    ȧ_4,α   =i(-ω_4,αa_4,α+λ_4,α a_1,α+ν_4,α a_2,α),
    .

α=1,…,L^2.





Fig. <ref> shows the densities of the bacteria at different instants as deduced by numerically integrating system (<ref>), and then by computing the mean values of the number operators over the initial state. We consider a nonhomogeneous region characterized by cell dependent parameters such that, when moving away from the center, the inertia parameters decrease and the interaction parameters moderately grow. Precisely, the parameters entering H are chosen as in <cit.>, say, ω_1,α=0.3/(1+d_α), ω_2,α=0.1/(1+d_α), ω_3,α=0.3/(1+d_α), ω_4,α=0.4/(1+d_α),
λ_2,α=0.5 d_α, λ_3,α=0.4d_α, λ_4,α=0.2d_α, ν_2,α=0.6d_α, ν_3,α=0.4d_α, provided that d_α≠ 0. Here  d_α is the Euclidean distance (normalized to 1) between the cell α and the central one; moreover,  μ_2,α=0.3 (α=1,…,L^2), whereas p_α,β=p_β,α is vanishing for α=β, equal to 1/d^2(α,β), where d(α,β) is the Euclidean distance between the cell α and the cell β belonging to the Moore neighborhood of cell α, and zero elsewhere. As initial data we chose a density 1  for the nutrients  over all the cells, a density 1 for the bacteria in the 9 central cells of the lattice, and  vanishing densities for the two garbages  all over the cells. These initial conditions reflect  the fact that, at t=0, the nutrients are uniformly distributed, the bacteria are localized in a restricted central area, and the two garbages are both empty. In Fig. <ref>, the mean and the variance of the densities of the various compartments all over the cells are shown. It can be seen that these average densities tend
to asymptotic values. The plot of variances shows a clear effect of the diffusion, that is a progressive
homogenization of the densities.

Once we have defined a suitable functional form for the Hamiltonian operator, we now move to describe the effect of combining the usual quantum definition of the dynamics with the action of specific rules periodically acting on the system. The rules we introduce have a simple physical or biological interpretation. As in Section <ref>, we will consider two different possibilities: the first one changes the vector of the system, while the second one acts on the parameters of the Hamiltonian.




 §.§ Rules acting on the state of the system

As previously stated in Section <ref>, if
φ_𝐧^0(0) is the initial state of the system,
the evolved state exp(-iHτ)φ_𝐧^0(0)  at time τ is mapped by the rule to a new state φ_𝐧^1∈ℋ, which becomes  the new initial state of the system for a new iteration, φ_𝐧^1.
By applying the rule several times, at equally spaced time steps, we produce a sequence of different initial states φ_𝐧^0(0),φ_𝐧^1(0),…, which evolve always with the same Hamiltonian H.
This approach was originally adopted in the Quantum Game of Life <cit.>.

Since our model involves 4 different populations
in L^2 cells, we construct the elements of the basis of the 2^4L^2-dimensional Hilbert space ℋ, spanned by

    φ_𝐧_1,𝐧_2,𝐧_3,𝐧_4=
         =(a^†_1,1)^n_1,1⋯(a^†_4,1)^n_4,1⋯(a^†_1,L^2)^n_1,L^2⋯(a^†_4,L^2)^n_4,L^2φ_0_1,0_2,0_3,0_4,

where φ_0_1,0_2,0_3,0_4 is the ground vector.
Here, 𝐧_𝐣 = (n_j,1,⋯,n_j,L^2) is the generic L^2-dimensional vector relative to the j-th population, and each n_j,α is 0 or 1. The initial state φ_𝐧^0(0) of the system can be now expressed as

    φ_𝐧^0(0)=∑_𝐧_1,𝐧_2,𝐧_3,𝐧_4c^0_𝐧_1,𝐧_2,𝐧_3,𝐧_4φ_𝐧_1,𝐧_2,𝐧_3,𝐧_4,

where c^0_𝐧_1,𝐧_2,𝐧_3,𝐧_4 are complex scalar. Although generally required in conservative quantum mechanics, there is no need to assume a normalized initial state, since, in general, it is not guaranteed, neither natural, that the rule we are adopting preserves the probabilistic interpretation of the states.

As an application to the bacterial population dynamics, we choose a rule such that, at the generic k-th iteration, the evolved state
φ_𝐧^(k-1)(τ)=exp(-iHτ)φ_𝐧^(k-1)(0) is mapped into a new state φ_𝐧^(k)(0) corresponding to removing the second garbage in some specific subregion 𝒪 of the lattice and reintroducing the same amount of garbage removed as nutrients for the bacteria, uniformly distributed all over the lattice.
More explicitly, at step k, we
compute the total amount of the garbage of the second type removed,

    N_g=∑_α∈𝒪n^(k-1)_4,α(τ),

and we set

    n^(k)_4,α(0)=0,   ∀α∈𝒪,
       n^(k)_1,α(0)=n^(k-1)_1,α(τ)+N_g/L^2,

if n^(k-1)_1,α(τ)+N_g/L^2≤1 in each cell α.
Otherwise, we remove only that part of the garbage which ensures the condition
n^(k-1)_1,α(τ)+N_g/L^2≤1 (for all α).

The densities of bacteria and of first garbage remain unchanged in each cell,

    n^(k)_2,α(0)=n^(k-1)_2,α(τ),  ∀α∈𝒪,
       n^(k)_2,α(0)=n^(k-1)_2,α(τ),  ∀α∈𝒪.


It is then evident that the sum of the densities stay constant,

    ∑_i,αn^(k)_i,α(0)=∑_i,α n^(k-1)_i,α(τ).

Hence, we define the new initial state φ_𝐧^(k)(0) fixing the coefficients in (<ref>) as

    c^k_ñ^α_1,0_2,0_3,0_4=√(n^k_1,α(0)),          c^k_0_1,ñ^α_2,0_3,0_4=√(n^k_2,α(0)),
       c^k_0_1,0_2,ñ^α_3,0_4=√(n^k_3,α(0)),          c^k_0_1,0_2,0_3,ñ^α_4=√(n^k_4,α(0)),

where 𝐧̃^α_j, for all j, is the α-th element of the canonical basis of ℝ^L^2, and 0_j is the zero vector in ℝ^L^2. It easy to check that the mean values of the various number operators n̂_j,α over the new initial state φ_𝐧^k(0) return exactly the densities n^(k)_j,α(0).

Hence, the rules ρ in (<ref>) which defines the new initial states at each new iterations are expressed through (<ref>)–(<ref>), and

    φ_𝐧^0(0) ρ⟶φ_𝐧^1(0) ρ⟶φ_𝐧^2(0) ρ⟶…ρ⟶φ_𝐧^k(0),


In analogy with (<ref>), we can construct the  sequence of density functions

    n̂_j,α(t)=
    {[    n^0_j,α(t)     t∈ [0,τ[,;  n^1_j,α(t-τ)   t∈ [τ,2τ[ ,; n^2_j,α(t-2τ) t∈ [2τ,3τ[  ,;             …               ]
    .

for each j and α.
The parameters entering H, as well as the initial data are those used previously, in absence of the rule.

Time evolutions of the mean values of the densities of the various populations together with their standard deviations over the lattice are shown in Figure <ref>, where we let the system evolve for 10 iterations, fixing τ=1 and L=27, and where the subregion 𝒪 consists in the two external rows/columns on the top-bottom and left-right of the lattice.  The plots show that the reintroduction of new nutrients in the system improves the overall density growth of all the populations in the lattice. This can be deduced since the mean values of the densities is here larger than in the case of absence of rules. In Fig. <ref> we show the densities of the bacteria and of the second garbage over all the lattice at times t=1.7, 3.6, 5.4; we notice  the presence of a gradient (more evident for the garbage) in the external part of the lattice. 


We end this section by observing that the rule considered here has a simple biological interpretation:  removing the hard garbage along the borders of the lattice, and introducing  new nutrients,  corresponds to a periodic cleaning of the system accompanied by a supplementary source of nutrients allowing the long term sustainability of the ecosystem.
This explains why the asymptotic mean densities over the lattice of the bacteria and of the two garbages are larger than the mean densities obtained for the problem without rules.










 §.§ Rules acting on the parameters of the model


Different rules can also be introduced to describe different biological effects, for instance changes in the metabolism of the bacteria due to stress factors. For instance, the increased density of the non recyclable garbage or the lack of nutrients are stress factors for the bacterial colonies, and cause changes in their behavior.

We start by using the values of the parameters and the initial conditions as stated above, and  take into account the dynamical mechanisms implied in each cell by the variation D_4,α of the scarcely recyclable garbage after any period of quantum evolution of the system of length τ,

    D_4,α=n_4,α(k τ)-n_4,α((k-1) τ),      k>1, α=1,…,L^2,

to generate a new set of values for some of the parameters of the model according to the rule ρ 

    [ [ ρ(ω_2,α)=ω_2,α(1+10 D_4,α),; ρ(λ_2,α)=λ_2,α(1-10 D_4,α),; ρ(λ_4,α)=λ_4,α(1-10 D_4,α),; ρ(ν_3,α)=ν_3,α(1-10 D_4,α). ] ]


Numerical simulations of the stepwise model describing the evolution of the system for consecutive time intervals in accordance with the Heisenberg representation and imposing repeatedly the rule ρ have been compared to those obtained by considering the evolution according to the standard 
nonhomogeneous linear model.
The initial data are those used in the previous sections.

As visible in Fig. <ref>, though the diffusion has the effect of distributing the bacteria all over the lattice (keep in mind that the radial inhomogeneity of the parameters determines the formation of symmetrical patterns), the validity of the proposed stepwise method, differently from what happens by adopting the standard linear model, is shown by the fact that the repeated application of the rule ρ after several time steps improves the description of the behavior of bacterial populations in terms of long term survival. Therefore, the proposed stepwise spatial method naturally describes  the relevant changes of the metabolism of certain genera, which are able to survive for a very long time in terms of latent life when under negative stimuli, without external actions.
The mean values and the variances of the densities of all the compartments of the ecosystem exhibit  an oscillating behavior with decreasing variations and the tendency to stabilize, in the case of the mean values, and to become smaller and smaller, in the case of the variances (see Fig. <ref> for the time evolutions corresponding to the simulation shown in Fig. <ref>).








§ CONCLUSIONS AND PERSPECTIVES


We have discussed how the notion of (H,ρ)-induced dynamics can be quite efficient to produce dynamical behaviors having some asymptotic non trivial limit, without the need of introducing any infinitely extended reservoir coupled to the system, or any explicit time dependent Hamiltonian. We have also shown that this is not really due to the explicit form of the rule ρ, but it is rather due to the presence itself of such a rule. In particular, we have compared different versions of the rule,  either changing the state of the system or modifying the parameters of the Hamiltonian of the system. Incidentally, this last choice could be seen as the use of a particularly simple time dependent Hamiltonian, which modifies itself along the time evolution. More precisely, it is more than this, as the results in the Appendix show. In fact, the application of the rule produces a piecewise constant quadratic Hamiltonian, whose related equations of motion can be solved quite easily, and then matched together. On the other hand, even for the simple example discussed in the Appendix, the choice of a time dependent Hamiltonian produces equations which, in general, are not so simple to solve, or can only be solved numerically. For this reason, our procedure appears more useful and technically much simpler. Moreover, as we have already discussed all along the paper, ρ has also a natural interpretation in terms of measures performed over the physical system, measures that, as it naturally happens in quantum mechanics, modify the system itself.

We should also mention that in the literature other ways are proposed to get some asymptotic behaviour for a given system. The approach based on master equation is one of this, <cit.>. This relies on the possibility of constructing a particularly simple dynamical equation, usually involving only few parameters. The model obtained in this way is easily solvable, but on the other hand several details of the interactions between the various agents of the original open system are lost. This is not what happens in our approach, which is much richer from this point of you. It is surely interesting, and is part of our future work, to compare in details these strategies. 






§ APPENDIX 1:  A TIME DEPENDENT HAMILTONIAN



As we have seen, the concept of (H,ρ)-induced dynamics can be efficiently used in the description of systems which are expected to reach some equilibrium, during their time evolution. Now we want to see if this can also be achieved without using any rule, but assuming that H can be explicitly dependent on time. To do this, we go back to the simple model considered in Section <ref>, and replace the Hamiltonian in (<ref>) by a similar, but time dependent, operator:

    H(t)=ω_1(t)a_1^† a_1+ω_2(t)a_2^† a_2 +λ(t)(a_1^† a_2+a_2^† a_1).

Here
ω_j(t) and λ(t) are real functions of the time. If we assume 𝒮 is initially in the state Ψ_0=∑_n_1,n_2=0^1 c_n_1,n_2φ_n_1,n_2, with ∑_n_1,n_2=0^1 |c_n_1,n_2|^2=1, the Schrödinger equation iΨ̇(t)=H(t)Ψ(t) can be rewritten, in terms of the time evolution of the coefficients c_n_1,n_2, as follows:

    i ċ_0,0(t)=0,
       i ċ_1,0(t)=ω_1(t) c_1,0(t) +λ(t)c_0,1(t),
       i ċ_0,1(t)=λ(t) c_1,0(t) +ω_2(t)c_0,1(t),
       i ċ_1,1(t)=(ω_1(t)  +ω_2(t))c_1,1(t),

with c_n_1,n_2(0)=c_n_1,n_2. It is clear that

    c_0,0(t)=c_0,0,      c_1,1(t)=c_1,1exp(-i(Ω_1(t)+Ω_2(t))),

where Ω_j(t)=∫_0^tω_j(t_1)dt_1, so that |c_0,0(t)|^2=|c_0,0|^2 and |c_1,1(t)|^2=|c_1,1|^2, for all t. Of course, the equations in (<ref>) suggest that the time evolution of c_0,1(t) and c_1,0(t) is decoupled from c_0,0(t) and c_1,1(t). However, this is not really so. In fact, since the time evolution of 𝒮 preserves the norm of Ψ(t), Ψ(t)=Ψ(0), we find that the coefficients c_n_1,n_2(t) must satisfy the constraint ∑_n_1,n_2=0^1 |c_n_1,n_2(t)|^2=1, which implies that

    |c_0,1(t)|^2+|c_1,0(t)|^2=1-|c_0,0(t)|^2-|c_1,1(t)|^2=1-|c_0,0|^2-|c_1,1|^2,

for all t. Extending what we have done in subsection <ref>, where we have computed n_j(t)=<φ_n_1,n_2,..
n̂_j(t)φ_n_1,n_2>, we are interested here in computing N_j^Ψ(t)=<Ψ(t),
n̂_jΨ(t)>. It turns out that

    N_1^Ψ(t)=|c_1,0(t)|^2+|c_1,1|^2,      N_2^Ψ(t)=|c_0,1(t)|^2+|c_1,1|^2,

which shows, in particular, that c_0,0 does not enter this computation, and that we do not need to follow the time evolution of c_1,1(t), since we only need its square modulus, which is constant. Concerning c_1,0(t) and c_0,1(t), if we introduce

    d_1,0(t)=c_1,0(t)e^iΩ_1(t),      d_0,1(t)=c_0,1(t)e^iΩ_2(t),

as well as Φ(t)=Ω_1(t)-Ω_2(t), we get the following system of coupled differential equations,

    i ḋ_1,0(t)= λ(t)e^iΦ(t)d_0,1(t),    
    i ḋ_0,1(t)=λ(t)e^-iΦ(t)d_1,0(t),

with initial conditions d_1,0(0)=c_1,0 and d_0,1(0)=c_0,1. Moreover N_1^Ψ(t)=|d_1,0(t)|^2+|c_1,1|^2 and  N_2^Ψ(t)=|d_0,1(t)|^2+|c_1,1|^2. It is clear that, if |c_0,0|^2+|c_1,1|^2=1, then both N_1^Ψ(t) and N_2^Ψ(t) admit an asymptotic value, which is N_j^Ψ(∞):=lim_t→∞N_j^Ψ(t)=|c_1,1|^2, j=1,2. This is a simple consequence of (<ref>), which implies that, in this case, d_1,0(t)=d_0,1(t)=0, for all t. From now on we will assume that c_0,1 or c_1,0, or both, are different from zero. Then, since

    d|d_1,0(t)|^2/dt=2iλ(t)[d_1,0(t)d_0,1(t)e^-iΦ(t)],

it is clear that |d_1,0(t)| converges to some asymptotic value (and N_1^Ψ(t) and N_2^Ψ(t) as well, as a  consequence), if either (i) λ(t) converges to zero when t diverges, or when (ii) lim_t→∞[d_1,0(t)d_0,1(t)e^-iΦ(t)]=0. The case (i) is completely clear: in absence of interaction the two densities have no way (and no reason) to change in time. On the other hand, the case (ii) is not particularly useful, since to check the condition we should solve the dynamical problem and compute  d_1,0(t) and d_0,1(t) first.

In order to find more cases which are "under control", we rewrite (<ref>) as a second order equation. Assuming that λ(t) is never zero, we find that

    d̈_1,0(t)-λ̇(t)+iλ(t)(ω_1(t)-ω_2(t))/λ(t)ḋ_1,0(t)+λ^2(t) d_1,0(t)=0,

with initial conditions d_1,0(0)=c_1,0 and ḋ_1,0(0)=-iλ(0)c_0,1. The equation for d_0,1(t) is simply the complex conjugate of equation (<ref>).

It is easy to find situations in which neither d_1,0(t) nor its modulus admits an asymptotic value. For instance, if we fix λ(t) in such a way λ̇(t)+iλ(t)(ω_1(t)-ω_2(t))=0, and we further restrict to ω_1(t)=ω_2(t), then d_1,0(t) is an oscillating function. The same conclusion is deduced if we try to keep in the equation (<ref>) the term proportional to ḋ_1,0(t), to have a sort of friction in the system. However, the only solution consistent with the fact that H(t)=H^†(t) is again an oscillating function. However, this does not imply that no solution d_1,0(t) exists such that |d_1,0(t)| admits an asymptotic value. This happens, in fact, if λ(t)=λ_0 is constant and |ω_1(t)-ω_2(t)| is a certain increasing function. For instance, in Figure <ref>(a) we plot |d_1,0(t)|^2 for ω_1(t)-ω_2(t)=-(exp(t)+1) and λ(t)=√(3). We are fixing here the initial conditions d_1,0(0)=1 and ḋ_1,0(0)=0. We see that, indeed, an asymptotic value is reached for |d_1,0(t)|^2, which means that also |d_0,1(t)|^2 converges to a limiting values when t diverges.



The same conclusion can be deduced considering Figure <ref>(b), where we plot again |d_1,0(t)|^2 for ω_1(t)-ω_2(t)=-t^2(exp(t)+1) and λ(t)=3. The initial conditions are here d_1,0(0)=.8 and ḋ_1,0(0)=1.

The plots we get here are qualitatively similar to those depicted in Fig. <ref>.
The conclusion of this analysis is that, as we have claimed several times along the paper, the use of the (H,ρ)–induced dynamics is not extremely different  from adopting a time dependent Hamiltonian.
In fact, when using the rule, the parameters of the Hamiltonian become step-wise constant functions. Moreover, parameters adjust themselves in a natural way leading to an equilibrium. On the other hand,   Hamiltonian H(t) in (<ref>) requires a very ad hoc choice of the functions to produce such an equilibrium. Last but not least, the use of the rule is simpler from a technical point of view.





§ ACKNOWLEDGEMENTS

The authors acknowledge partial support from GNFM of the INdAM. 
F. B. and F. G. acknowledge partial support from Palermo University. F.B. thanks the 
Distinguished Visitor Program of the Faculty of Science of the University of Cape Town, 2017.

99

bag2006 F. Bagarello, An operatorial approach to stock markets, J. Phys. A 39 (2006), 6823–6840.

bagbook F. Bagarello, Quantum dynamics for classical systems: with applications of the Number operator, J. Wiley and Sons, New York, 2012.

qdm1 A. Khrennikov, Ubiquitous quantum structure: from psychology to finances, Springer, Berlin, 2010.

qdm3 J. R. Busemeyer, P. D. Bruza, Quantum models of cognition and decision, Cambridge University Press, Cambridge, 2012.

qdm4 M. Asano, M. Ohya, Y. Tanaka, I. Basieva, A. Khrennikov, Quantum-like model of
brain's functioning: decision making from decoherence, J. Theor. Biol. 281 (2011), 56–64.

qdm5 M. Asano, M. Ohya, Y. Tanaka, I. Basieva, A. Khrennikov, Quantum-like dynamics of decision-making, Phys. A 391 (2012), 2083–2099.

Pkren1 P. Khrennikova, Modeling behavior of decision makers with the aid of algebra of 
qubit creation–annihilation operators, J.  Math. Psych. 78 (2017), 76-85.

BHK F. Bagarello, E. Haven, A. Khrennikov, A model of adaptive decision-making from representation of information environment by quantum fields, Phil. Trans. R. Soc. A 375 (2017),  20170162.

rev3c J. Broekaert, I. Basieva, P. Blasiak, E. Pothos, Quantum-like
dynamics applied to cognition: a consideration
of available options. Phil. Trans. R. Soc. A 375 (2017), 20160387. 

BO_migration F. Bagarello, F. Oliveri, An operator description of interactions between populations with applications to migration, Math. Mod. Meth. Appl. Sci. 23 (2013),  471–492.

BGO_crowds F. Bagarello, F. Gargano, F. Oliveri, A phenomenological operator description of dynamics of crowds: escape strategies,  
Appl. Math. Model. 39 (2015), 2276–2294.

GAR14 F. Gargano, Dynamics of Confined Crowd Modelled Using Fermionic Operators, Int. J. Theor. Phys., 53 (2014), 2727–2738.

BTBB F. Gargano, L. Tamburino, F. Bagarello, G. Bravo,
Large-scale effects of migration and conflict in pre-agricultural groups: Insights from a dynamic model, PloS ONE 12 (2017), e0172262.

BO_ecomod F. Bagarello, F. Oliveri, Dynamics of closed ecosystems described by operators, Ecol. Model. 275 (2014), 89–99.

BCO_desert F. Bagarello, A. M. Cherubini, F. Oliveri, An Operatorial Description of Desertification, SIAM J. Appl. Math. 76 (2016), 479–499.

DSO_RM2016 R. Di Salvo, F. Oliveri, An operatorial model for long–term survival of bacterial populations, Ricerche di Matematica  65 (2016), 435–447.

DSO_AAPP2016 R. Di Salvo, F. Oliveri, On fermionic models of a closed ecosystem with application to bacterial populations, Atti Accademia Peloritana dei Pericolanti 94, No. 2, A5 (2016), DOI:10.1478/AAPP.942A5.

pol1 P. Khrennikova, E. Haven, A. Khrennikov,, An application of the theory of open quantum systems to model the dynamics of party governance in the US Political System, Int. J. Theor. Phys. 53 (2014), 1346–1360.

pol4  M. Makowski, E. W. Piotrowski, Decisions in elections transitive or intransitive quantum preferences,  J. Phys. A: Math. Theor. 44 (2011), 215303.

all1 F. Bagarello, An operator view on alliances in politics, SIAM J. Appl. Math. 75 (2015),  564–584.

all2 F. Bagarello, E. Haven, First results on applying a non-linear effect formalism to
alliances between political parties and buy and sell dynamics, Physica A  444 (2016), 403–414.

all3  F. Bagarello, An improved model of alliances between political parties, Ricerche di Matematica 65 (2016), 399-412.

BG17 F. Bagarello,  F. Gargano,  Modeling interactions between political parties and electors, Physica A: Stat. Mech. Appl. 248 (2017), 243–264.

DSO_turncoat2017 R. Di Salvo, F. Oliveri,  An operatorial model for complex political system dynamics, Math. Meth. Appl. Sci. 40 (2017), 5668–5682.

DSGO_opinion2017 R. Di Salvo, M. Gorgone, F. Oliveri, (H, ρ)–induced political dynamics: facets of the disloyal attitudes Into the public opinion, Int. J. Theor. Phys. 56 (2017), 3912–3922.

DSGO_turncoat2017 R. Di Salvo, M. Gorgone, F. Oliveri, Political dynamics affected by turncoats,  Int. J. Theor. Phys.  56 (2017), 3604–3614.

rev3b P. Khrennikova, E. Haven, Instability of political
preferences and the role of mass media: a dynamical representation in a quantum framework, Phil. Trans. R. Soc. A 374 (2016), 20150106.

BDSGO_GoL F. Bagarello, R. Di Salvo,  F. Gargano, F. Oliveri, (H,ρ)-induced dynamics and the quantum game of life, Appl. Math. Model. 43 (2017),15–32.

Merzbacher E. Merzbacher, Quantum Mechanics, John Wiley and Sons, New York, 1998.

Messiah A. Messiah, Quantum mechanics, North Holland Publishing Company, Amsterdam, 1961.

romanP. Roman, Advanced quantum mechanics, Addison–Wesley, New York, 1965.

HHK2016 E. Haapasalo, T. Heinosaari, Y. Kuramochi, Saturation of repeated quantum measurements, J. Physics A: Math. Theor. 49 (2016) 33.

pascazio S. Pascazio, All you ever wanted to know about quantum Zeno effect in 70 minutes, Open. Sys. Inf. Dyn. 21 (2014), 1440007.

Moler C. Moler, C. Van Loan, Nineteen dubious ways to compute the exponential of a
matrix, Twenty-five years later, SIAM Rev. 45 (2003), 3–49.

Radmore S. M. Barnett, P. M. Radmore, Methods in theoretical quantum optics, Clarendon Press, Oxford, 1997.

tripf O. Cherbal, M. Drir, M. Maamache , D. A. Trifonov, 
Fermionic coherent states for pseudo-hermitian two-level systems, J. Phys. A, Math. Theor. 40 (2007), 1835–1844.

Givskov E. Givskov, L. Eberl, S. Molin, Responses to nutrient starvation in Pseudomonas putida KT2442: two-dimensional electrophoretic analysis of starvation- and stress-induced proteins, J. Bacteriology 176 (1994), 4816–4824.

Overbeek L. S. Van Overbeek, L. Eberl,  M. Givskov,  S. Molin, J. D. Van Elsas, Survival of, and induced stress resistance in, carbon-starved Pseudomonas fluorescens cells residing in soil, Appl. Env. Microbiology  61 (1995), 4202–4208.

Cabral J. P. S. Cabral, Viability and respiratory activity of Pseudomonas syringae cells starved in buffer, Can. J. Microbiology 41 (1995), 372–377.

Eberl L. Eberl, M. Givskov, C. Sternberg, S. Møller, G. Christiansen, S. Molin, Physiological responses of Pseudomonas putida KT2442 to phosphate starvation, Microbiology 142 (1996), 155–163.


Carnazza2007 S. Carnazza, S. Guglielmino, M. Nicolò, F. Santoro, F. Oliveri, A paradox in life thermodynamics: the long-term survival of bacterial populations?. Proceedings "WASCOM 2007", 14th Conference on Waves and Stability in Continuous Media. Ed. by N. Manganaro, 
R. Monaco, and S. Rionero. World Scientific, 135–140, 2008.

rev3a M. Asano, I. Basieva, A. Khrennikov,
M. Ohya, Y. Tanaka, I. Yamato, A model of epigenetic evolution based on theory of open quantum
systems, Syst. Synth. Biol. 7 (2003), 161–173.

Pkren2 P. Khrennikova, Application of quantum master equation for long-term
prognosis of asset-prices, Phys. A 450 (2016), 253–263.


