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The one-dimensional Galerkin-truncated Burgers equation, with both dissipation and noise terms included, is studied using spectral methods. When the truncation-scale Reynolds number R min is varied, from very small values to order 1 values, the scale-dependent correlation time τ ( k ) is shown to follow the expected crossover from the short-distance τ ( k ) ∼ k − 2 Edwards–Wilkinson scaling to the universal long-distance Kardar–Parisi–Zhang scaling τ ( k ) ∼ k − 3 / 2 . In the inviscid limit, R min → ∞ , we show that the system displays another crossover to the Galerkin-truncated inviscid-Burgers regime that admits thermalized solutions with τ ( k ) ∼ k − 1 . The scaling forms of the time-correlation functions are shown to follow the known analytical laws and the skewness and excess kurtosis of the interface increments distributions are characterized. This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.

We recall our approach through discretizations for path integrals and its general results for representations of probability densities. It is shown that the result of Arnold [P. Arnold, Phys. Rev. E 61, 6099 (2000)] is a particular case of our work.

The stationary probability is analytically determined in a suitable approximation for several systems presenting a weak noise transition. Numerical simulations are presented which corroborate the analytical results.

The normal form of a system making a Hopf bifurcation generically breaks rotational invariance z → z exp. We study the consequences of this effect on the corre-lation function and show that observable new peaks appear in the spectral density ∆(κ).

We study functional integrals in phase space and show that the definition without limiting procedure by a formal series expansion has the same ambiguities as the definition through discretization, these last ones being related to ordering problems. We exhibit the way to obtain the prescription that makes the series expansion unambiguous and study the mechanism that makes the expansion independent of the chosen discretization.

We show how to compute in a covariant was explicitly the higher order corrections in the semiclassical expansion of the free particle propagator in curved spaces.

Recently the perturbation theory for nonlinear systems was studied in detail by Leschke et al. using operator methods. We study and generalize their results in the functional integral formalism. Methods introduced in our previous work are generalized to include, in particular, normal and antinormal orderings of noncommuting operators.

Using the properties of the discretization techniques for functional integrals that we have developed before, we compute explicitly the higher-order corrections to the WKB approximation of the quantum system correspondingto the curved-space classical lagrangian L c 1 = 1 2 g μν q μ q ν +A μ q μ−V .

It results from recent works of Prigogine and collaborators that one can construct a nonunitary operator which realizes an “equivalence” between the positive actions of a reversible dynamical system and an irreversible Markov process going to equilibrium. We consider here this construction and we prove that (a) forK-shifts the transition probability of the associated Markov process is concentrated in the stable manifold of the transformed point by the shift with a point mass concentrated on the deterministic trajectory; and (b) for Bernoulli shifts the measures which go to equilibrium are the same for the deterministic system and the Markov process.

We study a method proposed recently that gives a unified view of the properties of stochastic fields determined by Langevin-type equations. The functional integrals involved are shown to need an additional prescription to be defined, this prescription being related to the relative order of noncommuting operators in the corresponding operator formalism which we characterize in detail. It is proved that the prescription is responsible for an additional term which has given rise to difficulties of interpretation in the Onsager-Machlup path probability density. We prove that all results are independent of this term. In particular in perturbation theory the cancellation mechanism of the prescription dependence is studied in detail.

The WKB expansion for an arbitrary Hamiltonian is computed by using functional integral techniques. All the higher-order corrections can be expressed in terms of one of the Green’s functions of the Jacobi operator of the associated classical problem.

We derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems. Using an appropriate scalar product in the space of homogeneous vector polynomials, we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator. This leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form. Explicit examples, and results obtained when there are additional symmetries, are also presented.

The Lagrangian which determines the most probable path for diffusion processes is obtained by a simple and intuitive technique based entirely on the use of path integrals.

We show how to compute in a covariant way the WKB-type expansion for the transition probability density of the Markovian processes generated by general multiplicative noise Langevin equations. The method uses phase space functional integrals and normal coordinates around the classical path. We compare with our earlier non-covariant methods and compute explicitly the first order correction to the WKB-approximation.

We give renormalization group representations for the propagator and four-point vertex function of thegϕ4-model. We state in a precise way the properties that are necessary for the existence of these representations in a global nonperturbative sense. Special attention is given to the invertibility postulate for the coupling constant which is explicitly discussed. We remark that in fact this property, as well as the renormalization group representations, just expresses the possibility of changing the parameters used to describe the theory (herem2 andg). The relation to the Callan-Symanzik equation is exhibited in view of a global discussion of the validity of this equation. An important case in which the global functionβ(g) vanishes identically is pointed out.

The steepest descent approximation is used in the functional integral representation of the solution of the Fokker-Planck equation. The calculation includes the Gaussian integrals of fluctuations around the classical solution. The problem of the prescription dependence of the functional integral is solved previously by transforming the representation to one involving a prescription independent functional integral.

Starting from the multivariate master equation we construct a reduced model that gives the critical theory. The relation to the static problem is studied using the minimal subtraction technique and the critical exponents are calculated up to O(ϵ 2 ).

We give a precise definition of the functional integrals involved by means of different discretization prescriptions. The value of the functional integral is prescription dependent and this leads to a variety of representations with different Lagrangians. We discuss general covariance and we derive in a unified way, all the lagrangians proposed in the literature together with their associated discretizations. We make also a critical examination of previous results.

It is shown how to compute explicitly and in a covariant way the higher-order corrections to the WKB approximation for propagators in curved spaces. We treat the quantum propagator corresponding to the classical system with Lagrangian L/sub cl/1/2(g-italic) = /sub munu/q/sup ..mu../q/sup ..nu../ + q/sup ..mu../A/sub ..mu../(q)-V(q). The method uses discrete techniques of phase-space functional integrals and involves normal coordinates around the classical path. The gauge invariance of the corrections is also explicitly exhibited.

We obtain the implications on the renormalization group representations of a renormalizable theory of the fact that the functionsβ(g) andγ(g) of the Callan-Symanzik equation are finite at every order of perturbation theory. We also study asymptotic behaviour and we conclude that it is impossible to decide from perturbation theory data if the behaviour for large momenta is independent or not of the coupling constantg. We explicitly exhibit, adopting an example of Jouvet, a mechanism leading to ag-dependent behaviour and which is consistent with all perturbation theory constraints. The situation is characterized by the global functionβ(g)≡0 although its perturbation series is different from zero.

We construct functional integral and operator formalisms for the stochastic process generated by a modified Langevin equation which contains a white noise and an independent markovian process taking discrete values.

The interaction of a classical field with general tensorial character with a particle described by its trajectory is shown to lead in general to a Lorentz-Dirac-type equation, independently of the details of the interaction. The divergences of the theory are carefully analysed by a method we have proposed before, which is shown to be equivalent to other regularization schemes that have been used previously. We apply our results to several well-known interactions including linearized gravitation and symmetric electrodynamics with monopoles.

We prove that if perturbation theory is reliable and if the propagator of the field satisfies the Kaellen-Lehmann representation then asymptotic freedom is equivalent to having Z/sub 3/(g ..-->.. 0) = 1 in the gphi/sup 4/ model. One may have Z/sub 3/(g) not-equal 0 and nonasymptotic freedom, but then Z/sub 3/(g ..-->.. 0) .. 0) .. 0) = 1 when Z/sub 3/(g) equivalent 0 and to give an anomalous-dimension function of the coupling constant when Z/sub 3/(g) equivalent 0.

We present the nonlinear phase equations describing the stability of a time-periodic one-dimensional spatial pattern, that arises in a system which is invariant by space and time translations and space reflection symmetry. We show that a large scale oscillatory instability can occur, leading to a quasiperiodic temporal regime with two different spatial scales On etablit les equations aux derivees partielles non lineaires qui gouvernent la stabilite a grande echelle d'une structure cellulaire unidimensionnelle oscillante, apparaissant dans un systeme hors equilibre, invariant par translations d'espace et de temps, et par reflexion d'espace. Existence d'une instabilite oscillatoire, conduisant a un regime quasi periodique, possedant deux echelles spatiales distinctes

A short derivation of the Feynman Lagrangian for general diffusion processes is given by a technique relying on the use of different discretisations which are related by equivalence relations under the n-dimensional integral whose limit is the path integral. In this way calculation of the differential equation satisfied by the path integral is avoided.

The critical behaviour of the Schlogl model is studied using field theoretical methods. It is shown that for the equal time correlation functions this behaviour is that of the well-known g ϕ 4 model.

We study a large class of quantum-field-theory models in a universe with no space dimension and one time dimension. We find continued-fraction expansions for the different matrix elements of our models and we are able to discuss in detail the convergence problems of the theories.

It is proved that the Z = 0 condition relates two theories with a different number of basic fields in such a way that the vacuum expectation values of T- products of fields are the same on-shell and off-shell in both theories, with the possible exception of the two-point function of the composite particle for which an HNZ local field is constructed in one of the theories. The connection to the Bethe- Salpeter method is also discussed together with other problems related to the Z = 0 condition.

We show that for a suitable definition of the vertex function of a composite particle in the sense ofZ3=0 we have the LSZ theorem. We also study the asymptotic behaviour of the form factor in this case.

The Hamiltonians of a class of models in quantum field theory are shown to be symmetric operators with deficiency index (1, 1). It follows that an infinite number of dynamical evolutions are possible. The connection between global methods of calculation and perturbation theory is studied for these models.

It is shown that critical values of the coupling constant which make the renormalization constant of a field vanish correspond to instability points of the solutions of the theory.

We study the probl]Literaturem of the existence of a lower bound for the Hamiltonians of a class of models in quantum field theory. We investigate the relation of this property to self-adjointness.

In a letter by Etim and Basili a thermodynamic lagrangian was constructed starting from the differential operator in the Kramers-Moyal expansion of the master equation. We clarify the relation between this approach and the definition of the path integral. This allows us to reinterpret some of the results.

We present an elementary and simple derivation of the Lorentz‐Dirac equation for a classical charged particle which clearly exhibits the origin of the mass renormalization one needs in this problem.