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D. H. E. Gross

Publications and source records attributed to D. H. E. Gross.

At least 55 records · Page 3Linked to original sources
arXiv2001

Non-extensive Hamiltonian systems follow Boltzmann's principle not Tsallis statistics. -- Phase Transitions, Second Law of Thermodynamics

Boltzmann's principle S(E,N,V)=k*ln W(E,N,V) relates the entropy to the geometric area e^{S(E,N,V)} of the manifold of constant energy in the N-body phase space. From the principle all thermodynamics and especially all phenomena of phase transitions and critical phenomena can be deduced. The topology of the curvature matrix C(E,N) (Hessian) of S(E,N) determines regions of pure phases, regions of phase separation, and (multi-)critical points and lines. Thus, C(E,N) describes all kind of phase-transitions with all their flavor. No assumptions of extensivity, concavity of S(E), or additivity have to be invoked. Thus Boltzmann's principle and not Tsallis statistics describes the equilibrium properties as well the approach to equilibrium of extensive and non-extensive Hamiltonian systems. No thermodynamic limit must be invoked.

cond-mat.stat-mech↗
arXiv2001

Effect of angular momentum on equilibrium properties of a self-gravitating system

The microcanonical properties of a two dimensional system of N classical particles interacting via a smoothed Newtonian potential as a function of the total energy E and the total angular momentum L are discussed. In order to estimate suitable observables a numerical method based on an importance sampling algorithm is presented. The entropy surface shows a negative specific heat region at fixed L for all L. Observables probing the average mass distribution are used to understand the link between thermostatistical properties and the spatial distribution of particles. In order to define a phase in non-extensive system we introduce a more general observable than the one proposed by Gross and Votyakov [Eur. Phys. J. B:15, 115 (2000)]: the sign of the largest eigenvalue of the entropy surface curvature. At large E the gravitational system is in a homogeneous gas phase. At low E there are several collapse phases; at L=0 there is a single cluster phase and for L>0 there are several phases with 2 clusters. All these pure phases are separated by first order phase transition regions. The signal of critical behaviour emerges at different points of the parameter space (E,L). We also discuss the ensemble introduced in a recent pre-print by Klinko & Miller; this ensemble is the canonical analogue of the one at constant energy and constant angular momentum. We show that a huge loss of informations appears if we treat the system as a function of intensive parameters: besides the known non-equivalence at first order phase transitions, there exit in the microcanonical ensemble some values of the temperature and the angular velocity for which the corresponding canonical ensemble does not exist, i.e. the partition sum diverges.

cond-mat.stat-mech↗
arXiv2001

Second Law of Thermodynamics, Macroscopic Observables within Boltzmann's Principle but without Thermodynamic Limit

Boltzmann's principle S=k ln W allows to extend equilibrium thermo-statistics to ``Small'' systems without invoking the thermodynamic limit. The clue is to base statistical probability on ensemble averaging and not on time averaging. It is argued that due to the incomplete information obtained by macroscopic measurements thermodynamics handles ensembles or finite-sized sub-manifolds in phase space and not single time-dependent trajectories. Therefore, ensemble averages are the natural objects of statistical probabilities. This is the physical origin of coarse-graining which is not anymore a mathematical ad hoc assumption. From this concept all equilibrium thermodynamics can be deduced quite naturally including the most sophisticated phenomena of phase transitions for ``Small'' systems. Boltzmann's principle is generalized to non-equilibrium Hamiltonian systems with possibly fractal distributions ${\cal{M}}$ in 6N-dim. phase space by replacing the conventional Riemann integral for the volume in phase space by its corresponding box-counting volume. This is equal to the volume of the closure $\bar{\cal{M}}$. With this extension the Second Law is derived without invoking the thermodynamic limit. The irreversibility in this approach is due to the replacement of the phase-space volume by the volume of its closure $\bar{\cal{M}}$. The physical reason for this replacement is that macroscopic measurements cannot distinguish ${\cal{M}}$ from $\bar{\cal{M}}$. Whereas the former is not changing in time due to Liouville's theorem, the volume of the closure can be larger. In contrast to conventional coarse graining the box-counting volume is defined in the limit of infinite resolution. I.e. there is no artificial loss of information.

cond-mat.stat-mech↗
arXiv2000

Second Law of Thermodynamics and Macroscopic Observables within Boltzmann's principle, an attempt

Boltzmann's principleS=k*ln W is generalized to non-equilibrium Hamiltonian systems with possibly fractal distributions in phase space by the box-counting volume. The probabilities P(M) of macroscopic observables M are given by the ratio P(M)=W(M)/W of these volumes of the sub-manifold {M} of the microcanonical ensemble with the constraint M to the one without. With this extension of the phase-space integral the Second Law is derived without invoking the thermodynamic limit. The irreversibility in this approach is due to the replacement of the phase space volume of the possibly fractal sub-manifold {M} by the volume of the closure of {M}. In contrast to conventional coarse graining the box-counting volume is defined by the limit of infinite resolution.

cond-mat.stat-mech↗
arXiv2000

What can nuclear collisions teach us about the boiling of water or the formation of multi-star systems ?

Phase transitions in nuclei, small atomic clusters and self-gravitating systems demand the extension of thermo-statistics to ``Small'' systems. The main obstacle is the thermodynamic limit. It is shown how the original definition of the entropy by Boltzmann as the volume of the energy-manifold of the N-body phase space allows a {\em geometrical} definition of the entropy as function of the conserved quantities. Without invoking the thermodynamic limit the whole ``zoo'' of phase transitions and critical points/lines can be unambiguously defined. The relation to the Yang--Lee singularities of the grand-canonical partition sum is pointed out. It is shown that just phase transitions in non-extensive systems give the complete set of characteristic parameters of the transition {\em including the surface tension.} Nuclear heavy-ion collisions are an experimental playground to explore this extension of thermo-statistics

cond-mat.stat-mech↗
arXiv2000

Phase Transitions in "Small" Systems - A Challenge for Thermodynamics

Traditionally, phase transitions are defined in the thermodynamic limit only. We propose a new formulation of equilibrium thermo-dynamics that is based entirely on mechanics and reflects just the {\em geometry and topology} of the N-body phase-space as function of the conserved quantities, energy, particle number and others. This allows to define thermo-statistics {\em without the use of the thermodynamic limit}, to apply it to ``Small'' systems as well and to define phase transitions unambiguously also there. ``Small'' systems are systems where the linear dimension is of the characteristic range of the interaction between the particles. Also astrophysical systems are ``Small'' in this sense. Boltzmann defines the entropy as the logarithm of the area $W(E,N)=e^{S(E,N)}$ of the surface in the mechanical N-body phase space at total energy E. The topology of S(E,N) or more precisely, of the curvature determinant $D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial N^2-(\partial^2S/\partial E\partial N)^2$ allows the classification of phase transitions {\em without taking the thermodynamic limit}. The topology gives further a simple and transparent definition of the {\em order parameter.} Attention: Boltzmann's entropy S(E) as defined here is different from the information entropy and can even be non-extensive and convex.

cond-mat.stat-mech↗
arXiv2000

Micro-canonical Statistical Mechanics of some Non-Extensive Systems

Non-extensive systems do not allow to go to the thermodynamic limit. Therefore we have to reformulate statistical mechanics without invoking the thermodynamical limit. I.e. we have to go back to Pre-Gibbsian times. We show that Boltzmann's mechanical definition of entropy S as function of the conserved ``extensive'' variables energy E, particle number N etc. allows to describe even the most sophisticated cases of phase transitions unambiguously for ``small'' systems like nuclei, atomic clusters, and selfgravitating astrophysical systems: The rich topology of the curvature of S(E,N) shows the whole ``Zoo'' of transitions: transitions of 1.order including the surface tension at phase-separation, continuous transitions, critical and multi-critical points. The transitions are the ``catastrophes'' of the Laplace transform from the ``extensive'' to the ``intensive'' variables. Moreover, this classification of phase transitions is much more natural than the Yang-Lee criterion.

cond-mat.stat-mech↗
arXiv1999

Phase Transitions in "Small" systems

Traditionally, phase transitions are defined in the thermodynamic limit only. We discuss how phase transitions of first order (with phase separation and surface tension), continuous transitions and (multi)-critical points can be seen and classified for small systems. Boltzmann defines the entropy as the logarithm of the area W(E,N)=e^S(E,N) of the surface in the mechanical N-body phase space at total energy E. The topology of the curvature determinant D(E,N) of S(E,N) allows the classification of phase transitions without taking the thermodynamic limit. The first calculation of the entire entropy surface S(E,N) for the diluted Potts model (ordinary (q=3)-Potts model plus vacancies) on a 50*50 square lattice is shown. The regions in {E,N} where D>0 correspond to pure phases, ordered resp. disordered, and D<0 represent transitions of first order with phase separation and ``surface tension''. These regions are bordered by a line with D=0. A line of continuous transitions starts at the critical point of the ordinary (q=3)-Potts model and runs down to a branching point P_m. Along this line \nabla D vanishes in the direction of the eigenvector v_1 of D with the largest eigen-value λ_1\approx 0. It characterizes a maximum of the largest eigenvalue λ_1. This corresponds to a critical line where the transition is continuous and the surface tension disappears. Here the neighboring phases are indistinguishable. The region where two or more lines with D=0 cross is the region of the (multi)-critical point. The micro-canonical ensemble allows to put these phenomena entirely on the level of mechanics.

cond-mat.stat-mech↗
arXiv1999

Phase transitions in finite systems = topological peculiarities of the microcanonical entropy surface

It is discussed how phase transitions of first order (with phase separation and surface tension), continuous transitions and (multi)-critical points can be defined and classified for finite systems from the topology of the energy surface $e^{S(E,N)}$ of the mechanical N-body phase space or more precisely of the curvature determinant $D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial N^2-(\partial^2S/\partial E\partial N)^2$ without taking the thermodynamic limit. The first calculation of the entire entropy surface $S(E,N)$ for a q=3-states Potts lattice gas on a 50*50 square lattice is shown. There are two lines, where $S(E,N)$ has a maximum curvature $\sim 0$. One is the border between the regions in \{$E,N$\} with $D(E,N)>0$ and with $D(E,N)<0$, the other line is critical starting as a valley in $D(E,N)$ running from the continuous transition in the ordinary q=3-Potts model, converting at $P_m$ into a flat ridge/plateau (maximum) deep inside the convex intruder of $S(E,N)$ which characterizes the first order liquid-gas transition. The multi-critical point $P_m$ is their crossing.

cond-mat.stat-mech↗
arXiv1999

Experimental and Theoretical Search for a Phase Transition in Nuclear Fragmentation

Phase transitions of small isolated systems are signaled by the shape of the caloric equation of state e^*(T), the relationship between the excitation energy per nucleon e^* and temperature. In this work we compare the experimentally deduced e^*(T) to the theoretical predictions. The experimentally accessible temperature was extracted from evaporation spectra from incomplete fusion reactions leading to residue nuclei. The experimental e^*(T) dependence exhibits the characteristic S-shape at e^* = 2-3 MeV/A. Such behavior is expected for a finite system at a phase transition. The observed dependence agrees with predictions of the MMMC-model, which simulates the total accessible phase-space of fragmentation.

nucl-th↗
arXiv1998

Phase Transitions without Thermodynamic Limit

Microcanonical statistics can be well applied to non-extensive systems like nuclei, atomic clusters and systems at phase transitions of first order with inhomogeneous configurations like phase separation. No thermodynamic limit has to be invoked. It allows to determine the transition temperature, latent heat and surface entropy. In the present paper this is demonstrated for systems of 200 - 3000 Na-, K-, and Fe-atoms at 1 atmosphere. As the energy is an exactly conserved quantity it is a usefull control parameter to tune through the development of the transition from the liquid side of the coexistence region over the evaporation, multifragmentation towards the final gas side. Microcanonical thermodynamics thus resolves details of the mechanism leading to the phase transition.

cond-mat↗
arXiv1998

Phase Transitions Without Thermodynamic Limit,The Crucial Rôle of Possible and Impossible Fluctuations, The Treatment of Inhomogeneous Scenaria in the Microcanonical Ensemble

Microcanonical thermodynamics (MCTh) is contrasted to canonical thermodynamics (CTh). At phase transitions of 1.order the two ensembles are NOT equivalent even in the thermodynamic limit . Energy fluctuations do not vanish and phase separations are suppressed in CTh. A proper treatment of fluctuations is neccessary. MCTh allows to address even isolated small systems where phase transitions can be clearly classified into first order and continuous ones. The microcanonical caloric curve T(E) determines the transition temperature, latent heat AND surface entropy/tension. For systems of ca.1000 Na-, K-, or Fe-atoms at 1 atm. all 3 quantities can be calculated. The three parameters approach with rising size the known bulk values. There is nothing that demands the use of the thermodynamic limit. Within microcanonical thermodynamics of finite systems there are fundamental differences between conserved extensive variables and ensemble related ones like entropy, temperature and pressure. This is discussed in detail.

cond-mat.stat-mech↗
arXiv1997

Fragmentation Phase Transition in Atomic Clusters IV -- Liquid-gas transition in finite metal clusters and in the bulk --

Within the micro-canonical ensemble phase transitions of first order can be identified without invoking the thermodynamic limit. We show for the liquid-gas transition of sodium, potassium, and iron at normal pressure that the transition temperature, the latent heat as well as the interface surface tension approach for some 1000 atoms the experimental bulk values. No excursion to the thermodynamic limit is neccessary. It is often easier to identify the transition in mesoscopic systems than in the infinite system. It is shown that the microcanonical T(E) contains important information about the transition which becomes suppressed in the canonical ensemble. The implications for our understanding of phase transitions in general are discussed. For these finite metallic systems it is essential to allow for a fragmentation of the system. Our result demonstrates further that microcanonical Metropolis Monte Carlo sampling (MMMC) is well suited to establish the link between the fragmentation phase transition in small systems like nuclei and atomic clusters and the liquid-gas transition of infinite matter.

cond-mat↗
arXiv1997

Fragmentation phase transition in atomic clusters I --- Microcanonical thermodynamics

Here we first develop the thermodynamics of microcanonical phase transitions of first and second order in systems which are thermodynamically stable in the sense of van Hove. We show how both kinds of phase transitions can unambiguously be identified in relatively small isolated systems of $\sim 100$ atoms by the shape of the microcanonical caloric equation of state $ $ I.e. within microcanonical thermodynamics one does not need to go to the thermodynamic limit in order to identify phase transitions. In contrast to ordinary (canonical) thermodynamics of the bulk microcanonical thermodynamics (MT) gives an insight into the coexistence region. The essential three parameters which identify the transition to be of first order, the transition temperature $T_{tr}$, the latent heat $q_{lat}$, and the interphase surface entropy $Δs_{surf}$ can very well be determined in relatively small systems like clusters by MT. The phase transition towards fragmentation is introduced. The general features of MT as applied to the fragmentation of atomic clusters are discussed. The similarities and differences to the boiling of macrosystems are pointed out.

cond-mat↗
arXiv1997

Fragmentation Phase Transition in Atomic Clusters II - Coulomb Explosion of Metal Clusters -

We discuss the role and the treatment of polarization effects in many-body systems of charged conducting clusters and apply this to the statistical fragmentation of Na-clusters. We see a first order microcanonical phase transition in the fragmentation of $Na^{Z+}_{70}$ for Z=0 to 8. We can distinguish two fragmentation phases, namely evaporation of large particles from a large residue and a complete decay into small fragments only. Charging the cluster shifts the transition to lower excitation energies and forces the transition to disappear for charges higher than Z=8. At very high charges the fragmentation phase transition no longer occurs because the cluster Coulomb-explodes into small fragments even at excitation energy $ε^* = 0$.

cond-mat↗
arXiv1997

Fragmentation Phase Transition in Atomic Clusters II -----Symmetry of Fission of Metal Clusters----

We present a statistical fragmentation study of doubly charged alkali (Li, Na, K) and antimony clusters. The evaporation of one charged trimer is the most dominant decay channel (asymmetric fission) at low excitation energies. For small sodium clusters this was quite early found in molecular dynamical calculations by Landman et al. For doubly charged lithium clusters, we predict Li$_{9}^{+}$ to be the preferential dissociation channel. As already seen experimentally a more symmetric fission is found for doubly charged antimony clusters. This different behavior compared to the alkali metal clusters is in our model essentially due to a larger fissility of antimony. This is checked by repeating the calculations for Na$_{52}^{++}$ with a bulk fissility parameter set artificially equal to the value of Sb.

cond-mat.mtrl-sci↗