The Mean Drift: Tailoring the Mean Field Theory of Markov Processes for Real-World Applications

The statement of the mean field approximation theorem in the mean field theory of Markov processes particularly targets the behaviour of population processes with an unbounded number of agents. However, in most real-world engineering applications one faces the problem of analysing middle-sized systems in which the number of agents is bounded. In this paper we build on previous work in this area and introduce the mean drift. We present the concept of population processes and the conditions under which the approximation theorems apply, and then show how the mean drift is derived through a systematic application of the propagation of chaos. We then use the mean drift to construct a new set of ordinary differential equations which address the analysis of population processes with an arbitrary size

Simplifying instanton corrections to N=4 SYM correlators

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Thermal quenches in N=2* plasmas

We exploit gauge/gravity duality to study `thermal quenches' in a plasma of the strongly coupled N=2* gauge theory. Specifically, we consider the response of an initial thermal equilibrium state of the theory under variations of the bosonic or fermionic mass, to leading order in m/T<<1. When the masses are made to vary in time, novel new counterterms must be introduced to renormalize the boundary theory. We consider transitions the conformal super-Yang-Mills theory to the mass deformed gauge theory and also the reverse transitions. By construction, these transitions are controlled by a characteristic time scale \calt and we show how the response of the system depends on the ratio of this time scale to the thermal time scale 1/T. The response shows interesting scaling behaviour both in the limit of fast quenches with T\calt<<1 and slow quenches with T\calt>>1. In the limit that T\calt\to\infty, we observe the expected adiabatic response. For fast quenches, the relaxation to the final equilibrium is controlled by the lowest quasinormal mode of the bulk scalar dual to the quenched operator. For slow quenches, the system relaxes with a (nearly) adiabatic response that is governed entirely by the late time profile of the mass. We describe new renormalization scheme ambiguities in defining gauge invariant observables for the theory with time dependant couplings.Comment: 78 pages, 17 figure

Finite-gap equations for strings on AdS_3 x S^3 x T^4 with mixed 3-form flux

We study superstrings on AdS_3 x S^3 x T^4 supported by a combination of Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz three form fluxes, and construct a set of finite-gap equations that describe the classical string spectrum. Using the recently proposed all-loop S-matrix we write down the all-loop Bethe ansatz equations for the massive sector. In the thermodynamic limit the Bethe ansatz reproduces the finite-gap equations. As part of this derivation we propose expressions for the leading order dressing phases. These phases differ from the well-known Arutyunov-Frolov-Staudacher phase that appears in the pure Ramond-Ramond case. We also consider the one-loop quantization of the algebraic curve and determine the one-loop corrections to the dressing phases. Finally we consider some classical string solutions including finite size giant magnons and circular strings.Comment: 44 pages, 3 figures. v2: references and a discussion about perturbative results adde

The a-theorem and conformal symmetry breaking in holographic RG flows

We study holographic models describing an RG flow between two fixed points driven by a relevant scalar operator. We show how to introduce a spurion field to restore Weyl invariance and compute the anomalous contribution to the generating functional in even dimensional theories. We find that the coefficient of the anomalous term is proportional to the difference of the conformal anomalies of the UV and IR fixed points, as expected from anomaly matching arguments in field theory. For any even dimensions the coefficient is positive as implied by the holographic a-theorem. For flows corresponding to spontaneous breaking of conformal invariance, we also compute the two-point functions of the energy-momentum tensor and the scalar operator and identify the dilaton mode. Surprisingly we find that in the simplest models with just one scalar field there is no dilaton pole in the two-point function of the scalar operator but a stronger singularity. We discuss the possible implications.Comment: 50 pages. v2: minor changes, added references, extended discussion. v3: we have clarified some of the calculations and assumptions, results unchanged. v4: published version in JHE

On correlation functions of operators dual to classical spinning string states

We explore how to compute, classically at strong coupling, correlation functions of local operators corresponding to classical spinning string states. The picture we obtain is of `fattened' Witten diagrams, the evaluation of which turns out to be surprisingly subtle and requires a modification of the naive classical action due to a necessary projection onto appropriate wave functions. We examine string solutions which compute the simplest case of a two-point function and reproduce the right scaling with the anomalous dimensions corresponding to the energies of the associated spinning string solutions. We also describe, under some simplifying assumptions, how the spacetime dependence of a conformal three-point correlation function arises in this setup.Comment: 27 pages, 3 figures; v2: references and comments added

Holographic three-point functions for short operators

We consider holographic three-point functions for operators dual to short string states at strong coupling in N=4 super Yang-Mills. We treat the states as point-like as they come in from the boundary but as strings in the interaction region in the bulk. The interaction position is determined by saddle point, which is equivalent to conservation of the canonical momentum for the interacting particles, and leads to conservation of their conformal charges. We further show that for large dimensions the rms size of the interaction region is small compared to the radius of curvature of the AdS space, but still large compared to the string Compton wave-length. Hence, one can approximate the string vertex operators as flat-space vertex operators with a definite momentum, which depends on the conformal and R-charges of the operator. We then argue that the string vertex operator dual to a primary operator is chosen by satisfying a twisted version of Q^L=Q^R, up to spurious terms. This leads to a unique choice for a scalar vertex operator with the appropriate charges at the first massive level. We then comment on some features of the corresponding three-point functions, including the application of these results to Konishi operators.Comment: 24 pages; v2: References added, typos fixed, minor change

Mineralisation of surfactants using ultrasound and the Advanced Fenton Process

The destruction of the surfactants, sodium dodecylbenzene sulfonate (DBS) and dodecyl pyridinium chloride (DPC), using an advanced oxidation process is described. The use of zero valent iron (ZVI) and hydrogen peroxide at pH = 2.5 (the advanced Fenton process), with and without, the application of 20 kHz ultrasound leads to extensive mineralisation of both materials as determined by total organic carbon (TOC)measurements. For DBS, merely stirring with ZVI and H2O2 at 20°C leads to a 51% decrease in TOC, but using 20 kHz ultrasound at 40°C, maintaining the pH at 2.5 throughout and adding extra amounts of ZVI and H2O2 during the degradation, then the extent of mineralisation of DBS is substantially increased to 93%. A similar result is seen for DPC where virtually no degradation occurs at 20°C, but if extra amounts of both ZVI and hydrogen peroxide are introduced during the reaction at 40°C and the pH is maintained at 2.5, then an 87% mineralisation of DPC is obtained. The slow latent remediation of both surfactants and the mechanism of degradation are also discussed

Matching three-point functions of BMN operators at weak and strong coupling

The agreement between string theory and field theory is demonstrated in the leading order by providing the first calculation of the correlator of three two-impurity BMN states with all non-zero momenta. The calculation is performed in two completely independent ways: in field theory by using the large-$N$ perturbative expansion, up to the terms subleading in finite-size, and in string theory by using the Dobashi-Yoneya 3-string vertex in the leading order of the Penrose expansion. The two results come out to be completely identical.Comment: 14 pages, 1 figur