 namely the large N approximation. Section III is devoted to the construction of the wave functional and a detailed description of the initial conditions for the problem. The dynamical aspects are covered in Section IV. We first outline some issues dealing with renormalization and then provide a qualitative understanding of the time evolution using wave functional arguments. We argue that the system could undergo dynamical symmetry breakdown and provide analytic estimates for the onset of instabilities. We present the results of our numerical calculations in Section V which confirm the robust features of the analytic estimates for a range of parameters. In Section VI we analyze the details of symmetry breaking and argue that the long time dynamics can be interpreted as the onset of formation of a Bose condensate even when the order parameter vanishes.
Finally in Section VII we present our conclusions and future avenues of study.

II. THE MODEL
As mentioned in the introduction we consider a λΦ 4 theory with O(N) symmetry in the large-N limit with the Lagrangian, where Φ is an O(N) vector, Φ = (σ, π) and π represents the N −1 pions, π = (π 1, π 2,..., π N −1 ). We then shift σ by its expectation value in the non-equilibrium state We refer the interested reader to several articles which discuss the implementation of the large N limit (see for e.g. [35][36][37][38][39]44,45]). The 1/N series may be generated by introducing an auxiliary field α(x) which is an algebraic function of Φ 2 (x), and then performing the functional integral over α(x) using the saddle point approximation in the large N limit [35][36][37]. It can be shown that the leading order terms in the expansion can be easily obtained by the following Hartree factorisation of the quantum fields [38,39,44,45], All expectation values are to be computed in the non-equilibrium state. In the leading order large N limit we then obtain, This approximation allows us to expand about field configurations that are far from the perturbative vacuum. In particular it is an excellent tool for studying the behaviour of matter in extreme conditions such as high temperature or high density [17,[35][36][37][38][39]44,45].
One way to obtain the non-equilibrium equations of motion is through the Schwinger-Keldysh Closed Time Path formalism. This is the usual Feynman path integral defined on a complex time contour which allows the computation of in-in expectation values as opposed to in-out S-matrix elements. For details see the references [40]. The Lagrangian density in this formalism is given by with the fields Φ ± defined along the forward (+) and backward (−) time branches. The non-equilibrium equations of motion are then obtained by requiring the expectation value of the quantum fluctuations in the non-equilibrium state to vanish i.e. from the tadpole equations [39], In the leading order approximation of the large N limit, the Lagrangian for the χ field is quadratic plus linear and the tadpole equation for the χ leads to the equation of motion for the order parameter or the zero modë In this leading approximation the non-equilibrium action for the N − 1 pions is, We have not written the action for the χ field fluctuations because they decouple from the dynamics of the pions in the leading order in the large N limit [38,39]. Having introduced the model and the non-perturbative approximation scheme the next step is to construct an appropriate non-equilibrium initial state or density matrix.
Although one could continue the analysis of the dynamics using the Schwinger-Keldysh method, we will study the dynamics in the Schrödinger representation in terms of wavefunctionals because this will display the nature of the quantum states more clearly. We find it convenient to work with the Fourier-transformed fields defined as, where we have chosen to quantize in a box of finite volume V that will be taken to infinity at the end of our calculations. The Hamiltonian for the pions is given by where is the effective time dependent frequency and M 2 π (t) is given by Eq.(2.5). To leading order in the large N limit the theory becomes Gaussian and the non-linearities are encoded in a self-consistency condition, since the frequency (2.15) depends on π 2 and this expectation value is in the time dependent state, as displayed by the set of equations (2.5-2.8).

III. THE INITIAL STATE
As stated in the introduction, our ultimate goal is to model and study the non-equilibrium aspects of the evolution of an initial, highly excited state that relaxes following high energy, heavy-ion collisions. An idealized description of the associated physics would be to consider two wave packets made up of very high energy components representing the heavy ions and moving with a highly relativistic momentum toward each other. The goal would be to follow the dynamical evolution of the wavefunctionals corresponding to this situation, thus clearly elucidating the non-equilibrium features involved in the phase transition processes following the interactions of the wave packets. This initial state could be described by a distribution of particles, sharply peaked around some special values k 0 and − k 0 in momentum space. The evolution of this state then follows from the functional Schrödinger equation.
Even with the simplification of a scalar field theory such a program is very ambitious and beyond the present numerical capabilities. One of the major difficulties is that selecting one particular momentum breaks rotational invariance and the evolution equations depend on the direction of wave vectors even in the Gaussian approximation. (This statement will become clear below).
In this article however, we choose to study a much simpler description of the initial state which is characterized by a high density particle distribution in a thin spherical'shell' in momentum space. We propose an initial particle distribution that has support concentrated at | k 0 |. This particular state does not provide the necessary geometry for a heavy ion collision, however it does describe a situation in which initially there is a large multiplicity of particles in a small momentum'shell', there is no special beam-axis and the pions are distributed equally in all directions with a sharp spatial momentum. This is a rotation invariant state that describes a highly out of equilibrium situation and that will relax during its time evolution (a spherical "tsunami").

A. The Wave Functional:
Since in the leading order approximation in the large N expansion the theory has become Gaussian (at the expense of a self-consistency condition), we choose a Gaussian ansatz for the wave-functional at t = 0. The reason for this choice is that upon time evolution this wave functional will remain Gaussian and will be identified with a squeezed state functional of pions.
This state will then evolve according to the Hamiltonian given in Eq. (2.14) which is essentially a harmonic oscillator Hamiltonian with self-consistent, time-dependent frequencies.
The functional Schrödinger equation is given by The last term in the Hamiltonian (2.14) which is independent of the fields (a time dependent 'vacuum energy term') can be absorbed in an overall time dependent phase of the wave functional. Removing this term by a phase redefinition, the functional Schrödinger equation becomes which then leads to a set of differential equations for the covariance A k. The time dependence of the normalization factors N k is completely determined by that of the A k as a consequence of unitary time evolution. The state for arbitrary time t takes then the form: The evolution equations for the covariance are obtained by taking the functional derivatives and comparing powers of η k on both sides. We obtain the following evolution equations [32,38] iȦ The equal time two-point correlation function in the time evolved non-equilibrium state is given by leading to the self-consistency condition.
Formally, one can also represent these two-point equal time correlators in terms of functional integrals over the closed time path contour where the initial state is chosen to be the Gaussian functional described above. However the explicit and rather simple ansatz for the wave functional enables one to obtain the two-point functions directly in a rather straightforward manner. Moreover, the wave functional approach will permit a much clearer understanding of the physics of the problem. The Ricatti equation (3.5) can be cast in a simpler form by writing A k in terms of new variables φ * k as leading to the simple equation for the new variables In terms of these mode functions we find that the real and imaginary parts of the covariance A k are given by From the differential equation for the φ k (t) given by Eq. (3.10) it is clear that the combination that appears in the numerator of Eq. (3.11) is the Wronskian Ω k of the differential equations and will consequently be determined from the initial conditions alone. The expression for the quantum fluctuations π 2 = π 2 /N is given by, The mode functions φ k have a very simple interpretation: they obey the Heisenberg equations of motion for the pion fields obtained from the Hamiltonian (2.14). Therefore we can write the Heisenberg field operators as where a k, a † k are the time independent annihilation and creation operators with the usual Bose commutation relations.

B. Initial Conditions:
Within this Gaussian ansatz for the wave functional, the initial conditions are completely determined by the initial conditions on the mode functions φ k (t). In order to physically motivate the initial conditions we now establish the relation between the particle number distribution and these mode functions.
Since, in a time dependent situation there is an ambiguity in the definition of the particle number, we define the particle number with respect to the eigenstates of the instantaneous Hamiltonian (2.14) at the initial time, i.e.
Here, ω k (0) is the frequency (2.15) evaluated at t = 0, i.e. the curvature of the potential term in the functional Schrödinger equation (3