Fluctuation theory in the Boltzmann--Grad limit

In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn (2003). We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process $\{\Phi(t)\}$ with transition kernel $P$ on a general state space $X$, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals $F$ on $X$, the kernel $\hat P(x,dy) = e^{F(x)} P(x,dy)$ has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a maximal, well-behaved solution to the multiplicative Poisson equation, defined as an eigenvalue problem for $\hat P$. Multiplicative Mean Ergodic Theorem: Consider the partial sums of this process with respect to any one of the functionals $F$ considered above. The normalized mean of their moment generating function (and not the logarithm of the mean) converges to the above maximal eigenfunction exponentially fast. Multiplicative regularity: The Lyapunov drift criterion under which our results are derived is equivalent to the existence of regeneration times with finite exponential moments for the above partial sums. Large Deviations: The sequence of empirical measures of the process satisfies a large deviations principle in a topology finer that the usual tau-topology, generated by the above class of functionals. The rate function of this LDP is the convex dual of logarithm of the above maximal eigenvalue, and it is shown to coincide with the Donsker-Varadhan rate function in terms of relative entropy. Exact Large Deviations Asymptotics: The above partial sums are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables.

Consider the normalized partial sums of a real-valued function F of a Markov chain, $$\\phi_{n}:=n^{-1}\\sum_{k=0}^{n-1}F(\\Phi(k)),\\qquad n\\ge1.$$ The chain {Φ(k):k≥0} takes values in a general state space $\\mathsf {X}$, with transition kernel P, and it is assumed that the Lyapunov drift condition holds: $PV\\le V-W+b\\mathbb{I}_{C}$ where $V: \\mathsf {X}\\to(0,\\infty)$, $W: \\mathsf {X}\\to[1,\\infty)$, the set C is small and W dominates F. Under these assumptions, the following conclusions are obtained:\n¶ 1. It is known that this drift condition is equivalent to the existence of a unique invariant distribution π satisfying π(W)<∞, and the law of large numbers holds for any function F dominated by W: $$ϕ_n→ϕ:=π(F),\\qquad\\mathrm{a.s.}, n→∞.$$\n¶ 2. The lower error probability defined by $\\mathsf {P}\\{\\phi_{n}\\le c\\}$, for c<ϕ, n≥1, satisfies a large deviation limit theorem when the function F satisfies a monotonicity condition. Under additional minor conditions an exact large deviations expansion is obtained.\n¶ 3. If W is near-monotone, then control-variates are constructed based on the Lyapunov function V, providing a pair of estimators that together satisfy nontrivial large asymptotics for the lower and upper error probabilities.\n¶ In an application to simulation of queues it is shown that exact large deviation asymptotics are possible even when the estimator does not satisfy a central limit theorem.

On the low density regime of homogeneous electron gas

SummarySomme rather general versions of the (so-called) Erdös-Rényi [14] law of large numbers have been derived by S. Csörgö [7] and Steinebach [21] for sequences of random variables satisfying a fist order large deviation theorem together with certain independence and stationarity properties. In view of recent second order large deviation asymptotics due to Dersch [13], corresponding refinements of the forementioned general Erdös-Rényi laws are discussed in this paper. The latter results can be viewed as convergence rate improvements of their earlier counterparts, typically providing the best rates, but not necessarily the best constants. Some specific examples are presented to demonstrate the applicability of our general approach.

The low density limit k0 to the rate coefficient for tertiary association of ions and polar molecules is related through a double integral to an entity z1 or z2 that involves only the temperature, the reduced mass of the colliding pair, the polarizability and dipole moment of the neutral, and k(exp)[M], where k(exp) is the measured rate coefficient at ambient gas density [M]. The results are parametrized so that k0 may easily be found from laboratory data (irrespective of whether or not third order kinetics prevail).

The dynamics of the localized region of the Lyapunov vector for the largest Lyapunov exponent is discussed in quasi-one-dimensional hard-disk systems at low density. We introduce a hopping rate to quantitatively describe the movement of the localized region of the Lyapunov vector, and show that it is a decreasing function of the hopping distance, implying a spatial correlation of the localized regions. This behavior is explained quantitatively by a brick accumulation model derived from hard-disk dynamics in the low density limit, in which hopping of the localized Lyapunov vector is represented as the movement of the highest brick position. We also give an analytical expression for the hopping rate, which is obtained as a sum of probability distributions for brick height configurations between two separated highest brick sites. The results of these simple models are in good agreement with the simulation results for hard-disk systems.

The shape of the melting line of the Wigner crystal is discussed in the temperature-density plane. It is argued that near the critical density ρ∗ for the Wigner transition at T = 0 the melting curve has the form For extreme low densities, on the other hand, the melting curve has as its asymptote the classical plasma melting line It seems that there should be a single maximum in the melting curve. The classical plasma line should provide an upper bound for the melting temperature, but is a poor bound away from the low density limit.

The family of elastically isotropic stretching-dominated cubic truss lattices

Two sets of related experiments are presented here. In the first, measurements of shear Alfven waves are used to test the predictions of a variety of different electron collision operators, including several Krook collision operators as well as a Lorentz collision operator. New expressions for the collisional warm-plasma dielectric tensor resulting from the use of the fully-magnetized collisional Boltzmann equation are presented here. Theoretical predictions for the parallel phase velocity and damping as a function of perpendicular wave number k⊥ are derived from the dielectric tensor. Laboratory measurements of the parallel phase velocity and damping of shear Alfven waves were made to test these theoretical predictions in both the kinetic (vte ≫ vA) and inertial (vte ≪ vA) parameter regimes and at several wave frequencies (ω < ωci). Results show that in the inertial regime, the best match between measurements and theory occur when any of the Krook operators are used to describe electron collisions. In contrast, the best agreement in the kinetic regime is found when collisions are completely ignored. In the second set of experiments, whistler waves were launched and received by a pair of dipole antennas immersed in the plasma at two positions along the background magnetic field. According to warm-plasma theory, there is absorbtion of the whistler wave when ω = |ωce| = eB/me due to resonance with the electrons. The whistler frequency was swept from somewhat below up to the electron cyclotron frequency |ωce|. As the frequency was swept, the wave was resonantly absorbed by those parts of electron phase space density which were Doppler shifted into resonance. The transmission of the wave through the plasma was measured. This measurement of transmission can be converted into a measure of the parallel electron distribution function. This diagnostic is designed to attempt to measure modifications to the parallel electron distribution function due to interactions with inertial Alfven waves. 2 Abstract Approved: Thesis Supervisor Title and Department

We study the dynamics of a homogeneous granular gas heated by a stochastic thermostat, in the low density limit. It is found that, before reaching the stationary regime, the system quickly "forgets" the initial condition and then evolves through a universal state that does not only depend on the dimensionless velocity, but also on the instantaneous temperature, suitably renormalized by its steady state value. We find excellent agreement between the theoretical predictions at the Boltzmann equation level for the one-particle distribution function and the direct Monte Carlo simulations. We conclude that at variance with the homogeneous cooling phenomenology, the velocity statistics should not be envisioned as a single-parameter, but as a two-parameter scaling form, keeping track of the distance to stationarity.

Εξετάζουμε προβλήματα πιθανοτήτων σχετικά με τη συμπεριφορά απλών γραμμικών στοχαστικών διαφορικών εξισώσεων με εκθετικά όρια, που σχετίζονται με προβλήματα που προκύπτουν στη θεωρία κινδύνου και στα μοντέλα asset και liability στα pension funds.Το πρώτο μοντέλο που εξετάζουμε, στο Κεφάλαιο 2, είναι μια διαδικασία Ornstein-Uhlenbeck (OU) που περιγράφεται από τη Στοχαστική Διαφορική Εξίσωση dX_t=μ X_t dt+σdW_t με X_0=x_0 δεδομένο, όπου μ&gt;0 και {W_t } είναι standard κίνηση Brown. Αυτό το μοντέλο προκύπτει ως μια προσέγγιση διάχυσης των μοντέλων θεωρίας κινδύνου στα οποία τα «ελεύθερα» αποθεματικά τοκίζονται. Το ερώτημα που τίθεται είναι αυτό του προσδιορισμού της πιθανότητας η διαδικασία να «χτυπήσει/συναντήσει» μια κάτω ντετερμινιστική καμπύλη υ_0 e^βt ή/και μια άνω καμπύλη u_0 e^αt, υποθέτοντας αρχικά ότι τα «ελεύθερα» αποθέματα βρίσκονται μεταξύ αυτών των τιμών, δηλ. 0&lt; υ_0&lt; x_0&lt; u_0 και ότι β&lt;μ&lt;α. Εξετάζονται τόσο το «πρόβλημα πιθανοτήτων καταστροφής» του πεπερασμένου ορίζοντα για τον προσδιορισμό της πιθανότητας πρόσκρουσης στο όριο εντός ενός πεπερασμένου ορίζοντα, όσο και η πιθανότητα άπειρου ορίζοντα. Αυτό το πρόβλημα μπορεί φυσικά να διατυπωθεί με όρους PDE δεύτερης τάξης με καμπύλα (εκθετικά) όρια στο επίπεδο και να λυθεί αριθμητικά. (Μια εναλλακτική προσέγγιση που περιλαμβάνει ένα επιχείρημα αλλαγής χρόνου (time change argument)αναφέρεται εν συντομία στο Κεφάλαιο 2.) Το κύριο μέρος της ανάλυσης περιλαμβάνει τεχνικές Large Deviations και ειδικότερα την προσέγγιση των Wentzell-Freidlin προκειμένου να ληφθούν λογαριθμικές ασυμπτωτικές για την πιθανότητα «πρόσκρουσης» είτε του κάτω είτε του άνω ορίου. Οι ασυμπτωτικές χαμηλού θορύβου ισχύουν όταν η διακύμανση σ είναι μικρή και ως εκ τούτου το γεγονός να χτυπήσει οποιοδήποτε όριο είναι σπάνιο. Ο εκθετικός ρυθμός που χαρακτηρίζει την πιθανότητα προκύπτει με την επίλυση ενός variational προβλήματος το οποίο δίνει επίσης το «μονοπάτι προς την καταστροφή». Ξεκινάμε με μια προσεκτική και λεπτομερή ανάλυση του προβλήματος του πεπερασμένου ορίζοντα της πρόσκρουσης σε ένα κατώτερο όριο. Το πρόβλημα του άπειρου ορίζοντα τόσο για το «χτύπημα» του κάτω όσο και του άνω εκθετικού ορίου αντιμετωπίζεται χρησιμοποιώντας την προσέγγιση των transversality conditions του λογισμού μεταβολών. Επιπλέον, εξετάζεται η διαδικασία OU με γενικότερο συντελεστή γραμμικής μετατόπισης, δηλαδή η διαδικασία που προκύπτει από την SDE dX_t=(μ X_t+r)dt+σdW_t με το άνω εκθετικό όριο u_0 e^αt (με 0&lt;μ&lt;α).Θεωρούμε επίσης, στο τέλος του Κεφαλαίου 2, το πρόβλημα δύο ανεξάρτητων OU διαδικασιών που προκύπτουν από τα SDEs dX_t=α X_t dt+σdW_t, dY_t=β Y_t dt+bdV_t, με δεδομένα X_0=x_0 , Y_0=y_0 . Επίσης, οι {W_t } και {V_t } είναι ανεξάρτητες κινήσεις Brown. Αν α&gt;β και x_0&gt; y_0 τότε, στην απουσία θορύβου, θα ισχύει ότι X_t&gt; Y_t για όλα τα t&gt;0. Εξετάζουμε, την πιθανότητα να συναντηθούν οι δύο διαδικασίες χρησιμοποιώντας ξανά την προσέγγιση των Wentzell-Freidlin. Οι βέλτιστες διαδρομές που ακολουθούνται από τις δύο διαδικασίες και ο χρόνος συνάντησης T καθορίζονται με την επίλυση ενός variational προβλήματος χρησιμοποιώντας transversality conditions. Είναι ενδιαφέρον ότι το ίδιο μοντέλο όταν θεωρήσουμε μια συσχέτιση μεταξύ των δύο κινήσεων Brown παρουσιάζει πιο περίπλοκη συμπεριφορά εάν ο συντελεστής συσχέτισης υπερβαίνει ένα ορισμένο όριο. Αυτή η τελευταία περίπτωση συζητείται στο κεφάλαιο 4.Στο Κεφάλαιο 3 εξετάζεται ένα αντίστοιχο πρόβλημα που περιλαμβάνει μια Γεωμετρική κίνηση Brown που περιγράφεται από τη SDE dX_t=μ X_t dt+σX_t dW_t με X_0=x_0 , μαζί με ένα άνω και ένα κάτω εκθετικό όριο. Και πάλι, χρησιμοποιείται η θεωρία των Wentzell-Freidlin. Σε αυτήν την περίπτωση, ωστόσο, μια ακριβής λύση είναι δυνατή, και επομένως μπορούμε να αποκτήσουμε μια ιδέα για την ακρίβεια των λογαριθμικών ασυμπτωτικών που προτείνουμε. Όπως αναμενόταν, όταν η διακύμανση σ γίνει μικρότερη, η εκτίμηση βελτιώνεται. Αναφέρεται επίσης η περίπτωση δύο συσχετιζόμενων γεωμετρικών κινήσεων Brown. Αυτά τα μοντέλα είναι εμπνευσμένα από το μοντέλο των Gerber και Shiu για τα assets &amp; liabilities στα pension funds.Στο Κεφάλαιο 4, εκτός από την επανεξέταση του προβλήματος δύο διαδικασιών Ornstein-Ulhenbeck με την παρουσία συσχέτισης, εξετάζουμε επίσης εν συντομία διαδικασίες OU με χρονικά μεταβαλλόμενη διακύμανση, που προκύπτουν από την SDE dX_t=μ X_t dt+σ(t)dW_t . Το πρόβλημα «πρόσκρουσης» που εξετάζουμε έχει κάτω εκθετικό όριο και άπειρο ορίζοντα. Το πρόβλημα μεταβλητότητας που προκύπτει από τη μέθοδο Wentzell-Freidlin είναι επιλύσιμο. Ωστόσο, η εξίσωση που δίνει τον βέλτιστο χρόνο «πρόσκρουσης» μπορεί να μην έχει μοναδική λύση. Επιλύουμε ένα παράδειγμα αυτού του προβλήματος αριθμητικά για να επεξηγήσουμε την προσέγγιση.

Consider a supercritical d-type branching process $Z_n^{i} = $ $ (Z_n^i(1), \cdots, Z_n^i(d)), \,n\geq 0,$ in an i.i.d. environment $\xi =(\xi_0, \xi_1, \ldots)$, starting with one particle of type $i$, whose offspring distributions of generation $n$ depend on the environment $\xi_n$ at time $n$. In the deterministic environment case, the famous Kesten-Stigum (1966) theorem states essentially that, if the mean matrix of the offspring distribution has spectral radius $\rho >1$, then for all $i,j =1, \ldots, d$, almost surely $ \lim_{n\rightarrow \infty} \frac{Z_{n}^{i}(j)}{\rho^n }$ exists and is finite; moreover, the limit variables are non-degenerate if and only if $\mathbb{E} \left(Z_1^{i}(j) \log^{+} Z_1^{i}(j)\right)<+\infty$ for all $i$ and $j$. The extension to the random environment case with $d=1$ has been done by Athreya and Karlin (1971) and Tanny (1988). Extending the Kesten-Stigum theorem to the random environment case with $d>1$ is a long-standing problem. The main objective of this paper is to resolve this delicate problem. In particular, under simple conditions, we prove that for any $1\le i,j\le d$, as $n \rightarrow +\infty$, $Z_n^{i}(j)/\mathbb{E}_\xi Z_n^{i}(j) \rightarrow W^{i}$ in probability, where $W^i$ is a non-negative random variable, $ \mathbb{E}_\xi Z_n^{i}(j)$ is the conditional expectation of $Z_n^{i}(j)$ given the environment $\xi$, which diverges to $\infty$ with geometric rate in the sense that $ \frac{1}{n} \log \mathbb{E}_\xi Z_n^{i}(j) \rightarrow \gamma >0$ almost surely, $\gamma$ being the Lyapunov exponent of the mean matrices of the offspring distributions; moreover $W^i$ are non-degenerate for all $i $ if and only if $\mathbb{E}\left(\frac{Z_1^{i}(j)}{M_0(i,j)}\log^{+}\frac{Z_1^{i}(j)}{M_0(i,j)}\right)<+\infty$ for all $i$ and $j$, where $M_0(i,j)$ is the conditioned mean of the number of children of type $j$ produced by a particle of type $i$ at time $0$, given the environment $\xi$. The key idea of the proof is the introduction of a non-negative martingale $(W_n^{i})$ which converges a.s. to $W^i$, and which reduces to the well-known fundamental martingale in the deterministic environment case. In addition, we prove that the direction $Z_n^{i}/ \| Z_n^{i} \|$ converges in law conditioned on the explosion event $\{\|Z_n^i\| \rightarrow +\infty\}$. The case of stationary and ergodic environment is also considered. Our results open ways to prove important properties such as central limit theorems with convergence rate and large deviation asymptotics.

An ideal gas is a particular molecular model system consisting of independent molecules without intermolecular forces and without volume. The molecules of an ideal gas are thus point particles, which are, when belonging to the same chemical compound, indistinguishable. Such a model system cannot exist in reality. However, in the limit of very low density the molecules of real a gas are, on the average, so far apart from each other that the forces between them and their volumes have no influence on some of the thermodynamic properties of the system. We may, therefore, expect that at least some aspects of thermodynamic behavior of real gases in the limit of low density are accurately described by the ideal gas model.

Pressure broadening as a many-body problem

Summary form only given. A dc electric field oriented in the growth direction of a quantum well (QW) changes the optical absorption and thus enables electrically driven optical modulators. Though the electroabsorption change is intrinsically very fast mechanism (since it does not involve change in carrier density), the device operation speed is ultimately limited by its electronic components. To increase the operation speed, the modulation by a terahertz (THz) electric field, such as provided by a free-electron laser or table-top sources, can be used. Such a THz field tuned at resonance with the intersubband transitions can induce large changes in the interband optical absorption. Moreover, the modulation by a THz field gives rise to optical emission at frequencies that differ from the frequency of the incident optical field by an integer number of the THz frequencies. This nonlinear optical/THz photon mixing has potential application in frequency shifting devices.Recently, we studied the optical properties of a THz driven QW resolving several subbands, both in the conduction and valence bands, using the polarization semiconductor Bloch equations in the low density limit.