Statistical Mechanics in the Extended Gaussian Ensemble

The extended gaussian ensemble (EGE) is introduced as a generalization of the canonical ensemble. The new ensemble is a further extension of the Gaussian ensemble introduced by J. H. Hetherington [J. Low Temp. Phys. {\bf 66}, 145 (1987)]. The statistical mechanical formalism is derived both from the analysis of the system attached to a finite reservoir and from the Maximum Statistical Entropy Principle. The probability of each microstate depends on two parameters $\beta$ and $\gamma$ which allow to fix, independently, the mean energy of the system and the energy fluctuations respectively. We establish the Legendre transform structure for the generalized thermodynamic potential and propose a stability criterion. We also compare the EGE probability distribution with the $q$-exponential distribution. As an example, an application to a system with few independent spins is presented.Comment: Revtex 4, 8 pages, 8 figure

Comparing the direct normal form method with harmonic balance and the method of multiple scales

Approximate analytical methods have been used extensively for finding approximate solutions to nonlinear ordinary differential equations. In this paper we compare the recently developed direct normal form transformation with two other very well known and long standing methods, harmonic balance and the method of multiple scales. We will show that the direct normal form method combines some of the key advantages of harmonic balance and multiple scales whilst reducing some of the limitations

An analytical approach for detecting isolated periodic solution branches in weakly nonlinear structures

AbstractThis paper considers isolated responses in nonlinear systems; both in terms of isolas in the forced responses, and isolated backbone curves (i.e. the unforced, undamped responses). As isolated responses are disconnected from other response branches, reliably predicting their existence poses a significant challenge. Firstly, it is shown that breaking the symmetry of a two-mass nonlinear oscillator can lead to the breaking of a bifurcation on the backbone curves, generating an isolated backbone. It is then shown how an energy-based, analytical method may be used to compute the points at which the forced responses cross the backbone curves at resonance, and how this may be used as a tool for finding isolas in the forced responses. This is firstly demonstrated for a symmetric system, where an isola envelops the secondary backbone curves, which emerge from a bifurcation. Next, an asymmetric configuration of the system is considered and it is shown how isolas may envelop a primary backbone curve, i.e. one that is connected directly to the zero-amplitude solution, as well as the isolated backbone curve. This is achieved by using the energy-based method to determine the relationship between the external forcing amplitude and the positions of the crossing points of the forced response. Along with predicting the existence of the isolas, this technique also reveals the nature of the responses, thus simplifying the process of finding isolas using numerical continuation

The Significance of Nonlinear Normal Modes for Forced Responses

Nonlinear normal modes (NNMs) describe the unforced and undamped periodic responses of nonlinear systems. NNMs have proven to be a valuable tool, and are widely used, for understanding the underlying behaviour of nonlinear systems. They provide insight into the types of behaviour that may be observed when a system is subjected to forcing and damping, which is ultimately of primary concern in many engineering applications. The definition of an NNM has seen a number of evolutions, and the contemporary definition encompasses all periodic responses of a conservative system. Such a broad definition is essential, as it allows for the wide variety of responses that nonlinear systems may exhibit. However, it may also lead to misleading results, as some of the NNMs of a system may represent behaviour that will only be observed under very specific forcing conditions, which may not be realisable in any practical scenario. In this paper, we investigate how the significance of NNMs may differ and how this significance may be quantified. This is achieved using an energy-based method, and is validated using numerical simulations

On the stability of the primordial closed string gas

We recast the study of a closed string gas in a toroidal container in the physical situation in which the single string density of states is independent of the volume because energy density is very high. This includes the gas for the well known Brandenberger-Vafa cosmological scenario. We describe the gas in the grandcanonical and microcanonical ensembles. In the microcanonical description, we find a result that clearly confronts the Brandenberger-Vafa calculation to get the specific heat of the system. The important point is that we use the same approach to the problem but a different regularization. By the way, we show that, in the complex temperature formalism, at the Hagedorn singularity, the analytic structure obtained from the so-called F-representation of the free energy coincides with the one computed using the S-representation.Comment: 20 pages and 1 figure. The final version that appeared in JHE

An analytical method for the optimisation of weakly nonlinear systems

In this paper we discuss how backbone curves can be used to guide the design and optimisation of weakly nonlinear systems with multiple degrees-of-freedom. Aft er decomposing the system using the modes of the equivalent linear system (the linear modes), we show how the backbone curves of the unforced, undamped equivalent system can be calculated. These consist of pure responses in each of the linear modes and, in certain parameter regimes, responses which are a combination of two or more linear modes - a feature which can be linked to internal resonance. Using an example system we will investigate how these backbone curves can be used to describe particular characteristics of the response. An energy balancing technique is also employed to relate the backbone curves to the response of the forced and damped system, and anticipate the conditions for which a particular characteristic will be seen. Finally, we discuss how the analytical nature of these techniques enables us to precisely design and optimise characteristics of such systems and how this can be expanded to systems with a greater number of degrees-of-freedom

Football in the community schemes: Exploring the effectiveness of an intervention in promoting healthful behaviour change

This study aims to examine the effectiveness of a Premier League football club’s Football in the Community (FitC) schemes intervention in promoting positive healthful behaviour change in children. Specifically, exploring the effectiveness of this intervention from the perspectives of the participants involved (i.e. the researcher, teachers, children and coaches). A range of data collection techniques were utilized including the principles of ethnography (i.e. immersion, engagement and observations), alongside conducting focus groups with the children. The results allude to the intervention merely ‘keeping active children active’ via (mostly) fun, football sessions. Results highlight the important contribution the ‘coach’ plays in the effectiveness of the intervention. Results relating to working practice (i.e. coaching practice and coach recruitment) are discussed and highlighted as areas to be addressed. FitC schemes appear to require a process of positive organizational change to increase their effectiveness in strategically attending to the health agenda

Mayer and virial series at low temperature

We analyze the Mayer pressure-activity and virial pressure-density series for a classical system of particles in continuous configuration space at low temperature. Particles interact via a finite range potential with an attractive tail. We propose physical interpretations of the Mayer and virial series' radius of convergence, valid independently of the question of phase transition: the Mayer radius corresponds to a fast increase from very small to finite density, and the virial radius corresponds to a cross-over from monatomic to polyatomic gas. Our results have consequences for the search of a low density, low temperature solid-gas phase transition, consistent with the Lee-Yang theorem for lattice gases and with the continuum Widom-Rowlinson model.Comment: 36 pages, 1 figur