Local thermodynamics and the generalized Gibbs-Duhem equation in systems with long-range interactions

The local thermodynamics of a system with long-range interactions in d dimensions is studied using the mean-field approximation. Long-range interactions are introduced through pair interaction potentials that decay as a power law in the interparticle distance. We compute the local entropy, Helmholtz free energy, and grand potential per particle in the microcanonical, canonical, and grand canonical ensembles, respectively. From the local entropy per particle we obtain the local equation of state of the system by using the condition of local thermodynamic equilibrium. This local equation of state has the form of the ideal gas equation of state, but with the density depending on the potential characterizing long-range interactions. By volume integration of the relation between the different thermodynamic potentials at the local level, we find the corresponding equation satisfied by the potentials at the global level. It is shown that the potential energy enters as a thermodynamic variable that modifies the global thermodynamic potentials. As a result, we find a generalized Gibbs-Duhem equation that relates the potential energy to the temperature, pressure, and chemical potential. For the marginal case where the power of the decaying interaction potential is equal to the dimension of the space, the usual Gibbs-Duhem equation is recovered. As examples of the application of this equation, we consider spatially uniform interaction potentials and the self-gravitating gas. We also point out a close relationship with the thermodynamics of small systems.

A Preliminary Review of Behavioural Biometrics for Health Monitoring in the Elderly

This article explores the potential of ICT-based biometrics for monitoring the health status of the elderly people. It departs from specific ageing and biometric traits to then focus on behavioural biometric traits like handwriting, speech and gait to finally explore their practical application in health monitoring of elderly.

Nucleus accumbens is involved in human action monitoring: evidence from invasive electrophysiological recordings

The Nucleus accumbens (Nacc) has been proposed to act as a limbic-motor interface. Here, using invasive intraoperative recordings in an awake patient suffering from obsessive-compulsive disease (OCD), we demonstrate that its activity is modulated by the quality of performance of the subject in a choice reaction time task designed to tap action monitoring processes. Action monitoring, that is, error detection and correction, is thought to be supported by a system involving the dopaminergic midbrain, the basal ganglia, and the medial prefrontal cortex. In surface electrophysiological recordings, action monitoring is indexed by an error-related negativity (ERN) appearing time-locked to the erroneous responses and emanating from the medial frontal cortex. In preoperative scalp recordings the patient's ERN was found to be signifi cantly increased compared to a large (n = 83) normal sample, suggesting enhanced action monitoring processes. Intraoperatively, error-related modulations were obtained from the Nacc but not from a site 5 mm above. Importantly, crosscorrelation analysis showed that error-related activity in the Nacc preceded surface activity by 40 ms. We propose that the Nacc is involved in action monitoring, possibly by using error signals from the dopaminergic midbrain to adjust the relative impact of limbic and prefrontal inputs on frontal control systems in order to optimize goal-directed behavior.