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In this paper we use spontaneous flux production in annular superconductors to shed light on the Kibble-Zurek scenario. In particular, we examine the effects of finite size and external fields, neither of which is directly amenable to the KZ analysis. Supported by 1D and 3D simulations, the properties of a superconducting ring are seen to be well represented by analytic Gaussian approximations which encode the KZ scales indirectly. Experimental results for annuli in the presence of external fields corroborate these findings., Comment: 20 pages, 10 figures; submitted to J. Phys: Condens. Matter for the special issue 'Condensed Matter Analogues of Cosmology'; v2: considerably reduced length, incorporation of experimental details into main text, discussion improved, references added, version accepted for publication

Consistent with the predictions of Kibble and Zurek, scaling behaviour has been seen in the production of fluxoids during temperature quenches of superconducting rings. However, deviations from the canonical behaviour arise because of finite-size effects and stray external fields. Technical developments, including laser heating and the use of long Josephson tunnel junctions, have improved the quality of data that can be obtained. With new experiments in mind we perform large-scale 3D simulations of quenches of small, thin rings of various geometries with fully dynamical electromagnetic fields, at nonzero externally applied magnetic flux. We find that the outcomes are, in practice, indistinguishable from those of much simpler Gaussian analytical approximations in which the rings are treated as one-dimensional systems and the magnetic field fluctuation-free., Comment: 10 pages, 3 figures, presentation at QFS2012, to appear in JLTP

In this article we examine a variety of quantitative models for describing archaeological networks, with particular emphasis on the maritime networks of the Aegean Middle Bronze Age. In particular, we discriminate between those gravitational networks that are most likely (maximum entropy) and most efficient (best cost/benefit outcomes)., Comment: 21 pages, 6 figures, 2 tables. Contribution to special issue of Advances in Complex Systems from the conference `Cultural Evolution in Spatially Structured Populations', UCL, London, September 2010. To appear in Advances in Complex Systems

We report on the first experimental verification of the Zurek-Kibble scenario in an isolated superconducting ring over a wide parameter range. The probability of creating a single flux quantum spontaneously during the fast normal-superconducting phase transition of a wide Nb loop clearly follows an allometric dependence on the quenching time $\tau_{Q}$, as one would expect if the transition took place as fast as causality permits. However, the observed Zurek-Kibble scaling exponent $\sigma = 0.62\pm0.15$ is two times larger than anticipated for large loops. Assuming Gaussian winding number densities we show that this doubling is well-founded for small annuli., Comment: 10 pages, 2 figures, submitted to Phys. Rev. Letts

In the context of quasi-Hermitian theories, which are non-Hermitian in the conventional sense, but can be made Hermitian by the introduction of a dynamically-determined metric $\eta$, we address the problem of how the functional integral and the Feynman diagrams deduced therefrom "know" about the metric. Our investigation is triggered by a result of Bender, Chen and Milton, who calculated perturbatively the one-point function $G_1$ for the quantum Hamiltonian $H=\half(p^2+x^2)+igx^3$. It turns out that this calculation indeed corresponds to an expectation value in the ground state evaluated with the $\eta$ metric. The resolution of the problem turns out be that, although there is no explicit mention of the metric in the path integral or Feynman diagrams, their derivation is based fundamentally on the Heisenberg equations of motion, which only take their standard form when matrix elements are evaluated with the inclusion of $\eta$.

We investigate the Hagedorn transitions of string networks with Y-junctions as may occur, for example, with (p,q) cosmic superstrings. In a simplified model with three different types of string, the partition function reduces to three generalised coupled XY models. We calculate the phase diagram and show that, as the system is heated, the lightest strings first undergo the Hagedorn transition despite the junctions. There is then a second, higher, critical temperature above which infinite strings of all tensions, and junctions, exist. Conversely, on cooling to low temperatures, only the lightest strings remain, but they collapse into small loops.

We report on the spontaneous production of fluxons in the presence of a symmetry-breaking magnetic field for annular Josephson tunnel junctions during a thermal quench. The dependence on field intensity $B$ of the probability $\bar{f_1}$ to trap a single defect during the N-S phase transition drastically depends on the sample circumferences. We show that the data can be understood in the framework of the Kibble-Zurek picture of spontaneous defect formation controlled by causal bounds., Comment: Submitted to Phys. Rev. B with 5 figures on Nov. 15, 2007

In this further letter on the onset of classical behaviour in field theory due to a phase transition, we show that it can be phrased easily in terms of the decoherence functional, without having to use the master equation. To demonstrate this, we consider the decohering effects due to the displacement of domain boundaries, with implications for the displacement of defects, in general. We see that decoherence arises so quickly in this event, that it is negligible in comparison to decoherence due to field fluctuations in the way defined in our previous papers., Comment: Version published in Phys. Lett. B

In the Schroedinger formulation of non-Hermitian quantum theories a positive-definite metric operator $\eta\equiv e^{-Q}$ must be introduced in order to ensure their probabilistic interpretation. This operator also gives an equivalent Hermitian theory, by means of a similarity transformation. If, however, quantum mechanics is formulated in terms of functional integrals, we show that the $Q$ operator makes only a subliminal appearance and is not needed for the calculation of expectation values. Instead, the relation to the Hermitian theory is encoded via the external source $j(t)$. These points are illustrated and amplified for two non-Hermitian quantum theories: the Swanson model, a non-Hermitian transform of the simple harmonic oscillator, and the wrong-sign quartic oscillator, which has been shown to be equivalent to a conventional asymmetric quartic oscillator., Comment: 14 pages, no figures

It can be shown using operator techniques that the non-Hermitian $PT$-symmetric quantum mechanical Hamiltonian with a "wrong-sign" quartic potential $-gx^4$ is equivalent to a Hermitian Hamiltonian with a positive quartic potential together with a linear term. A naive derivation of the same result in the path-integral approach misses this linear term. In a recent paper by Bender et al. it was pointed out that this term was in the nature of a parity anomaly and a more careful, discretized treatment of the path integral appeared to reproduce it successfully. However, on re-examination of this derivation we find that a yet more careful treatment is necessary, keeping terms that were ignored in that paper. An alternative, much simpler derivation is given using the additional potential that has been shown to appear whenever a change of variables to curvilinear coordinates is made in a functional integral., Comment: LaTeX, 12 pages, no figures

It has been argued by Zurek and Kibble that the likelihood of producing defects in a continuous phase transition depends in a characteristic way on the quench rate. In this paper we discuss an improved experiment for measuring the Zurek-Kibble scaling exponent $\sigma $ for the production of fluxons in annular symmetric Josephson Tunnel Junctions. We find $\sigma \simeq 0.5$. Further, we report accurate measurements of the junction gap voltage temperature dependence which allow for precise monitoring of the fast temperature variations during the quench., Comment: 12 pages, 5 figures, submitted to Phys. Rev. B

Some quantum field theories described by non-Hermitian Hamiltonians are investigated. It is shown that for the case of a free fermion field theory with a $\gamma_5$ mass term the Hamiltonian is $\cal PT$-symmetric. Depending on the mass parameter this symmetry may be either broken or unbroken. When the $\cal PT$ symmetry is unbroken, the spectrum of the quantum field theory is real. For the $\cal PT$-symmetric version of the massive Thirring model in two-dimensional space-time, which is dual to the $\cal PT$-symmetric scalar Sine-Gordon model, an exact construction of the $\cal C$ operator is given. It is shown that the $\cal PT$-symmetric massive Thirring and Sine-Gordon models are equivalent to the conventional Hermitian massive Thirring and Sine-Gordon models with appropriately shifted masses., Comment: 9 pages, 1 figure

New scaling behavior has been both predicted and observed in the spontaneous production of fluxons in quenched $Nb-Al/Al_{ox}/Nb$ annular Josephson tunnel junctions as a function of the quench time, $\tau_{Q}$. The probability $f_{1}$ to trap a single defect during the N-S phase transition clearly follows an allometric dependence on $\tau_{Q}$ with a scaling exponent $\sigma = 0.5$, as predicted from the Zurek-Kibble mechanism for {\it realistic} JTJs formed by strongly coupled superconductors. This definitive experiment replaces one reported by us earlier, in which an idealised model was used that predicted $\sigma = 0.25$, commensurate with the then much poorer data. Our experiment remains the only condensed matter experiment to date to have measured a scaling exponent with any reliability., Comment: Four pages, one figure

If systems change as fast as possible as they pass through a phase transition then the initial domain structure is constrained by causality. We shall show how we can trace these causal horizons by measuring the spontaneous production of flux in annular Josephson Tunnel Junctions as a function of the quench time $\tau_{Q}$ into the superconducting phase. A specific test of our analysis is that the probability P$_{1}$ to trap a single fluxon at the N-S transition clearly follows an allometric dependence on $\tau_{Q}$ as $P_1 = a \tau_Q^{-\sigma}$, with a scaling exponent $\sigma = 0.25$, in agreement with the data., Comment: Based on talks presented at the COSLAB meeting at Ambleside, 2004, to be published in 'Progress in Josephson Junction Research, ed. F. Columbus (Nova Science Publishers, Inc., USA), 2005

We continue the analysis of the onset of classical behaviour in a scalar field after a continuous phase transition, in which the system-field, the long wavelength order parameter of the model, interacts with an environment, of its own short-wavelength modes and other fields, neutral and charged, with which it is expected to interact. We compute the decoherence time for the system-field modes from the master equation and directly from the decoherence functional (with identical results). In simple circumstances the order parameter field is classical by the time the transition is complete., Comment: 10 pages, 1 figure: To be published in the International Journal of Theoretical Physics (2005) as part of the Proceedings of the "Peyresq Physics 9" meeting (2004) on "Micro and Macro structures of spacetime",ed. E. Verdaguer

In these lectures we look for parallels between symmetry breaking in the early universe and condensed matter systems, and discuss experiments that display these., Comment: 13 pages, published in the Proceedings of the National Workshop on Cosmological Phase Transitions and Topological Defects (Porto, Portugal, 2003), ed. T.A. Girard (Grafitese, Edificio Ciencia, 2004), pp11-23

We analyze the onset of classical behaviour in a scalar field after a continuous phase transition, in which the system-field, the long wavelength order parameter of the model, interacts with an environment of its own short-wavelength modes. We compute the decoherence time for the system-field modes from the master equation and compare it with the other time scales of the model. Within our approximations the decoherence time is in general the smallest dynamical time scale. Demanding diagonalisation of the decoherence functional produces identical results. The inclusion of other environmental fields makes diagonalisation occur even earlier., Comment: Seven pages, no figures. Contributed talk to the Second International Workshop DICE2004, Piombino, Italy. To be published in the Brazilian Journal of Physics

A recent experiment that sees the spontaneous creation of magnetic flux on quenching high-$T_c$ films has shown that earlier null results were a consequence of the lack of saturation of the Zurek-Kibble causal bounds against which flux density was measured. In this letter we estimate how efficient the production of topological charge is in planar systems, both for the aforementioned experiment (when flux measures topological charge) and for an earlier experiment on planar liquid crystals. Agreement is good., Comment: 4 pages, no figures

It has been argued by Zurek and Kibble that the likelihood of producing defects in a continuous phase transition depends in a characteristic way on the quench rate. In this paper we discuss our experiment for measuring the Zurek-Kibble scaling exponent $\sigma $ for the production of fluxons in annular symmetric Josephson Tunnel Junctions. The predicted exponent is $% \sigma =0.25$, and we find $\sigma =0.27\pm 0.05$. Further, there is agreement with the ZK prediction for the overall normalisation., Comment: 10 pages, 5 figures: To be published in Patterns of Symmetry Breaking (Ed. Jacek Dziarmaga), Proceedings of the NATO Advanced Study Institute & COSLAB School, Krakow, Poland (September 2002) Proceedings of The Nato Advanced Study Institute and COSLAB School, September 2002, Krakow, Poland

Classical defects (monopoles, vortices, etc.) are a characteristic consequence of many phase transitions of quantum fields. We show a model in which the onset of classical probability distributions, for the long-wavelength modes at early times, allows the identification of line-zeroes of the field with vortex separation. We obtain a refined version of Kibble's causal results for defect separation, but from a completely different approach. It is apparent that vortices are not created from thermal fluctuations in the Ginzburg regime., Comment: 10 pages, RevTex file. No figures. To appear in Phys. Lett. B

The conductivity of a finite temperature 1+1 dimensional fermion gas described by the massive Thirring model is shown to be related to the retarded propagator of the dual boson sine-Gordon model. Duality provides a natural resummation which resolves infra-red problems, and the boson propagator can be related to the fermion gas at non-zero temperature and chemical potential or density. In addition, at high temperatures, we can apply a dimensional reduction technique to find resummed closed expressions for the boson self-energy and relate them to the fermion conductivity. Particular attention is paid to the discussion of analytic continuation. The resummation implicit in duality provides a powerful alternative to the standard diagrammatic evaluation of transport coefficients at finite temperature., Comment: 41 pages, 6 figures

We study the onset of a classical order parameter after a second-order phase transition in quantum field theory. We consider a quantum scalar field theory in which the system-field (long-wavelength modes), interacts with its environment, represented both by a set of scalar fields and by its own short-wavelength modes. We compute the decoherence times for the system-field modes and compare them with the other time scales of the model. We analyze different couplings between the system and the environment for slow quenches. Within our approximations decoherence is in general a short time event., Comment: 17 pages, no figures; RevTex file. Many important changes. Version to appear in Nuclear Physics B

Classical defects (monopoles, vortices, etc.) are a characteristic consequence of many phase transitions of quantum fields. Most likely these include transitions in the early universe and such defects would be expected to be present in the universe today. We continue our analysis of the onset of classical behaviour after a second-order phase transition in quantum field theory and show how defects appear after such transitions, Comment: 10 pages, no figures, to be published in the Proceedings of the Peyresq VI, Cosmological inflation and primordial fluctuations. Energy desert and submillimeter gravity, Peyresq, June 2001

We analyze the onset of classical field configurations after a phase transition. Firstly, we motivate the problem by means of a toy model in quantum mechanics. Subsequently, we consider a scalar field theory in which the system-field interacts with its environment, represented both by further scalar fields and by its own short-wavelength modes. We show that, for very rapid quenches, the order parameter can be treated classically by the time that it has achieved its ground state values (spinodal time)., Comment: 12 pages, 8 figures, to be published in the Proceedings of the Peyresq VI, Cosmological inflation and primordial fluctuations. Energy desert and submillimeter gravity, Peyresq, June 2001

Phase transitions executed in a finite time show a domain structure with defects, that has been argued by Zurek and Kibble to depend in a characteristic way on the quench rate. In this letter we present an experiment to measure the Zurek-Kibble scaling exponent sigma. Using symmetric and long Josephson Tunnel Junctions, for which the predicted index is sigma = 0.25, we find sigma = 0.27 +/- 0.05. Further, there is agreement with the ZK prediction for the overall normalisation., Comment: To be published in Phys. Rev. Letts

Zurek's and Kibble's causal constraints for defect production at continuous transitions are encoded in the field equations that condensed matter systems and quantum fields satisfy. In this article we highlight some of the properties of the solutions to the equations and show to what extent they support the original ideas., Comment: To be published in Journal of Low Temperature Physics, June/July 2001 as a general (refereed) article in the proceedings of the ULTI conference, Finland, 2001

We analyze the onset of classical behaviour after a second-order phase transition by considering a scalar field theory in which the system-field interacts with its environment, represented both by further fields and by its own short-wavelength modes. Within our approximations we see that the long-wavelength modes have become classical by the time that the transition has been first implemented (the spinodal time)., Comment: 12 pages, final version to appear in Phys. Lett. B

The dual 'worldline', or 'string', description of adiabatic Ginzburg-Landau field theory near a phase transition in terms of quasi-Brownian strings and loops is well understood. In reality, the implementation of a transition is intrinsically non-equilibrium. We sketch how time-dependent Ginzburg-Landau theory leads to a modified dual string picture in which a causal bound on the growth of unstable string prevents the uncontrolled proliferation of string (the Shockley-Hagedorn transition) suggested by the adiabatic approximation., Comment: 4 pages Revtex, no figures

We focus on the massive Thirring model in 1+1 dimensions at finite temperature and non-zero chemical potential, and comment on some parallels between this model and QCD. In QCD, calculations of physical quantities such as transport coefficients are extremely difficult. In the massive Thirring model, similar calculations are greatly simplified by exploiting the duality which exists with the sine-Gordon model and its relation, at high temperature, to the exactly solvable classical Coulomb gas on the line., Comment: 5 pages, 1 figure, talk given by D.A.Steer at Strong and Electro-Weak Matter 2000 (SEWM 2000), Marseille, 14 - 17 June 2000

In parallel with Kibble's description of the onset of phase transitions in the early universe, Zurek has provided a simple picture for the onset of phase transitions in condensed matter systems, strongly supported by agreement with experiments in He3. In this letter we show how experiments with annular Josephson tunnel Junctions can and do provide further support for this scenario., Comment: Revised version with correct formula for the Swihart velocity. The results are qualitatively the same as with the previous version but differ quantitatively. 4 pages, RevTex

Kibble and Zurek have provided a unifying causal picture for the appearance of classical defects like cosmic strings or vortices at the onset of phase transitions in relativistic QFT and condensed matter systems respectively. In condensed matter the predictions are partially supported by agreement with experiments in superfluid helium. We provide an alternative picture for the initial appearance of defects that supports the experimental evidence. When the original predictions fail, this is understood, in part, as a consequence of thermal fluctuations (noise), which play a comparable role in both condensed matter and QFT., Comment: 25 pages, to be published in the Proceedings of the Fourth Conference on Quantum and Stochastic Gravity, String Cosmology and Inflation, Peyresq, September, 1999

Kibble and Zurek have provided a unifying causal picture for the appearance of topological defects like cosmic strings or vortices at the onset of phase transitions in relativistic QFT and condensed matter systems respectively. There is no direct experimental evidence in QFT, but in condensed matter the predictions are largely, but not wholly, supported in superfluid experiments on liquid helium. We provide an alternative picture for the initial appearance of strings/vortices that is commensurate with all the experimental evidence from condensed matter and consider some of its implications for QFT., Comment: 37 pages, to be published in Condensed Matter Physics, 2000

We examine the Zurek scenario for the production of vortices in quenches of liquid $^{4}He$ in the light of recent experiments. Extending our previous results to later times, we argue that short wavelength thermal fluctuations make vortices poorly defined until after the transition has occurred. Further, if and when vortices appear, it is plausible that that they will decay faster than anticipated from turbulence experiments, irrespective of quench rates., Comment: 4 pages, Revtex file, no figures Apart from a more appropriate title, this paper differs from its predecessor by including temperature, as well as pressure, quenches

We show that in certain limits the (1+1)-dimensional massive Thirring model at finite temperature $T$ is equivalent to a one-dimensional Coulomb gas of charged particles at the same $T$. This equivalence is then used to explore the phase structure of the massive Thirring model. For strong coupling and $T>>m$ (the fermion mass) the system is shown to behave as a free gas of "molecules" (charge pairs in the Coulomb gas terminology) made of pairs of chiral condensates. This binding of chiral condensates is responsible for the restoration of chiral symmetry as $T\to\infty$. In addition, when a fermion chemical potential $\mu\neq 0$ is included, the analogy with a Coulomb gas still holds with $\mu$ playing the role of a purely imaginary external electric field. For small $T$ and $\mu$ we find a typical massive Fermi gas behaviour for the fermion density, whereas for large $\mu$ it shows chiral restoration by means of a vanishing effective fermion mass. Some similarities with the chiral properties of low-energy QCD at finite $T$ and baryon chemical potential are discussed., Comment: 28 pages, 6 figures, better resolution figures are available upon request

Zurek has provided a simple picture for the onset of the lamda-transition in He4, not currently supported by vortex density experiments. However, we argue that the seemingly similar argument by Zurek that superflow in an annulus of He4 at the quench will be measurable is still valid., Comment: 4 pages with 1 figure

Kibble and Zurek have provided a unifying picture for the onset of phase transitions in relativistic QFT and condensed matter systems respectively, strongly supported by agreement with condensed matter experiments in He3. The failure of a recent experiment on He4 to agree with the predictions has prompted a reappraisal of this picture. We provide an alternative picture that explains the experimental evidence., Comment: Talk presented at 5th International Workshop on Thermal Field Theories and Their Applications, Regensberg, August 1998. 10 pages, no figures

We study bosonisation in the massive Thirring and sine-Gordon models at finite temperature and nonzero fermion chemical potential. Both canonical operator and path integral approaches are used to prove the equality of the partition functions of the two models at finite $T$ and zero chemical potential, as it has been recently shown. This enables the relationship between thermal normal ordering and path-integral renormalisation to be specified. Furthermore, we prove that thermal averages of zero-charge operators can also be identified. At nonzero chemical potential and temperature we show, in perturbation theory around the massless case, that the bosonised theory is the sine-Gordon model plus an additional topological term, accounting for the existence of zero charge excitations (the fermions or the kinks) in the thermal bath. This result is the 2D version of the low-energy lagrangian at finite baryon density., Comment: 10 pages RevTex, Proceedings of the 5th International Workshop on Thermal Field Theories and their application, Regensburg, Germany, August 10-14, 1998

In the light of recent difficulties in observing vortices in quenches of liquid helium 4 to its superfluid state we re-examine the Zurek scenario for their production. We argue that experiments in helium 4 are unlikely to produce true vortices in the numbers originally anticipated, if at all, because of the wide Ginzberg regime and the slowness of the mechanical quenches. On the other hand, the observed production of unambiguous vortices in neutron-bombarded helium 3, with its narrow Ginzberg regime, and fast quenches, is to be expected., Comment: RevTeX, 4 pages

In the light of recent difficulties in observing vortices in quenches of liquid $^{4}$He to its superfluid state we re-examine the Zurek scenario for their production. We find that the standard prediction is suspect because of a confusion over correlation lengths., Comment: 4 pages, Revtex

We calculate the initial density of relativistic global vortices (strings) produced at a quench and contrast it with the predictions of Kibble and Zurek., Comment: Latex file, 8 pages, No figures. The text contains minor changes that elucidate the onset of classical behaviour more thoroughly. Also including the word temperature in the title makes the nature of the transition more explicit

We present a model for the formation of relativistic global vortices (strings) at a quench, and calculate their density and correlations. The significance of these results to early universe and condensed-matter physics is discussed. In particular, there is always open, or infinite, string., Comment: uuencoded postscript file, 41 pages. Latex and postscript versions also available at http://euclid.tp.ph.ic.ac.uk/Papers/index.html

We present a model for the formation of relativistic vortices (strings) at a quench, and calculate their density and correlations. The significance of these to early universe and condensed-matter physics is discussed., Comment: 12 pages, LaTeX (no figures), available through anonymous ftp as LaTeX from ftp://euclid.tp.ph.ic.ac.uk/papers/94-5_47.tex and on WWW as LaTeX or postscript at http://euclid.tp.ph.ic.ac.uk/Papers/index.html

We examine string (vortex) formation at a quench for a weakly-coupled global U(1) theory when the excitation spectrum is non-relativistic. It is so similar to vortex production in the corresponding relativistic plasma as to reinforce arguments for the similarity of vorheptex production in the early universe and in low-temperature many-body physics., Comment: 23 pages, replaced version - tex problems corrected

We examine string (vortex) formation at a quench for a weakly-coupled global U(1) theory when the excitation spectrum is non-relativistic. It is so similar to vortex production in the corresponding relativistic plasma as to reinforce arguments for the similarity of vortex production in the early universe and in low-temperature many-body physics., Comment: 20 pages, RevTeX and epsf, 1 postscript figure. Resubmitted because archive (not a bulletin board) ate the figures file first time round. Available through anonymous ftp as LaTeX from ftp://euclid.tp.ph.ic.ac.uk/papers/94-5_53.tex or as LaTeX or postscript at http://euclid.tp.ph.ic.ac.uk/Papers/index.html

We develop winding number correlation functions that allow us to assess the role of field fluctuations on vortex formation in an Abelian gauge theory. We compute the behavior of these correlation functions in simple circumstances and show how fluctuations are important in the vicinity of the phase transition. We further show that, in our approximation, the emerging population of long/infinite string is produced by the classical dynamics of the fields alone, being essentially unaffected by field fluctuations., Comment: Latex file, 27 pages. 8 figures, available in compressed form by anonymous ftp from ftp://euclid.tp.ph.ic.ac.uk/papers/94-5_39.fig Latex and postscript versions also available at http://euclid.tp.ph.ic.ac.uk/Papers/index.html

We shall show that the density of defects produced at a second-order phase transition is determined by the correlation length of the fields. This is true both for defects appearing in the Ginzburg regime and for defects produced at a quench, when the Ginzburg regime is irrelevant. (Invited lectures given by R.J.R. to the Nato Advanced Study Institute and Euroconference on Formation and Interaction of Topological Defects, Isaac Newton Institute, Cambridge (UK), September, 1994.), Comment: 44 pages, LaTeX (figures included as LaTeX), available through anonymous ftp from ftp://euclid.tp.ph.ic.ac.uk/papers/94-5_06.tex or at http://euclid.tp.ph.ic.ac.uk/Papers/index.html

Our understanding of the mechanism by which topological defects are formed in symmetry breaking phase transitions has recently changed. We examine the non-equilibrium dynamics of defect formation for weakly-coupled global O(N) theories possessing vortices (strings) and monopoles. It is seen that, as domains form and grow, defects are swept along on their boundaries at a density of about one defect per coherence area (strings) or per coherence volume (monopoles)., Comment: 16 pages

We use the closed time-path formalism to calculate fluctuations at phase transitions, both in and out of equilibrium. Specifically, we consider the creation of vortices by fluctuations, of relevance to the early universe and to $^{4}He$ superfluidity. (Invited talk to Nato Advanced Research Workshop on Electroweak Physics and the Early Universe, Sintra (Portugal), March 1994), Comment: 17 pages, LaTeX, Imperial/TP/93-94/39

We derive analytic expressions for the interaction energy between two general $U(1)$ cosmic strings as the function of their relative orientation and the ratio of the coupling constants in the model. The results are relevant to the statistic description of strings away from critical coupling and shed some light on the mechanisms involved in string formation and the evolution of string networks., Comment: 31 pages,REVTEX, Imperial/TP/93-94/33

We construct a new coarse-grained effective potential which enables us to estimate the probabilities of thermal fluctuations above an arbitrary threshold at different length scales., Comment: (LaTeX, 25pp plus 2 figs, available from mark@amtp.cam.ac.uk) DAMTP-93-28, Imperial/TP/92-93/40