Showing papers on "Boltzmann constant published in 2023"

Unifying semiclassics and quantum perturbation theory at nonlinear order

Abstract: Nonlinear electrical response permits a unique window into effects of band structure geometry. It can be calculated either starting from a Boltzmann approach for small frequencies, or using Kubo's formula for resonances at finite frequency. However, a precise connection between both approaches has not been established. Focusing on the second order nonlinear response, here we show how the semiclassical limit can be recovered from perturbation theory in the velocity gauge, provided that finite quasiparticle lifetimes are taken into account. We find that matrix elements related to the band geometry combine in this limit to produce the semiclassical nonlinear conductivity. We demonstrate the power of the new formalism by deriving a quantum contribution to the nonlinear conductivity which is of order $\tau^{-1}$ in the relaxation time $\tau$, which is principally inaccessible within the Boltzmann approach. We outline which steps can be generalized to higher orders in the applied perturbation, and comment about potential experimental signatures of our results.

The Boltzmann distributions of folded molecular structures predict likely changes through random mutations

Abstract: New folded molecular structures can only evolve after arising through mutations. This aspect is modelled using genotype-phenotype (GP) maps, which connect sequence changes through mutations to changes in molecular structures. Previous work has shown that the likelihood of appearing through mutations can differ by orders of magnitude from structure to structure and that this can affect the outcomes of evolutionary processes. Thus, we focus on the phenotypic mutation probabilities ϕ qp, i.e. the likelihood that a random mutation changes structure p into structure q. For both RNA secondary structures and the HP protein model, we show that a simple biophysical principle can explain and predict how this likelihood depends on the new structure q: ϕ qp is high if sequences that fold into p as the minimum-free-energy structure are likely to have q as an alternative structure with high Boltzmann frequency. This generalises the existing concept of plastogenetic congruence from individual sequences to the entire neutral spaces of structures. Our result helps us understand why some structural changes are more likely than others, can be used as a basis for estimating these likelihoods via sampling and makes a connection to alternative structures with high Boltzmann frequency, which could be relevant in evolutionary processes.

A Quasi-Statistical Approach to the Boltzmann Entropy Equation Based on a Novel Energy Conservation Principle

Abstract: Boltzmann entropy equation is gained according to the statistical mechanics directly and general dependence between entropy and probability is obtained. Based on the second law of thermodynamics with a glance at the Boltzmann entropy equation, it can be deduced that physical processes are done in a direction that the probability of the system and total entropy increase. In fact, the possible process performing states and their entropy variations will be determined at a specific energy level. In this paper, an entropy equation is gained by using a new quasi-statistical approach to the physical processes as well as a novel energy conservation principle. The variation of the "energy structure equation”, as an equation to formulate the performed process using activated energy components of the system and their dependence, is studied in different possible paths by using the energy conservation principle directly. Despite the classical mechanics that all particles are studied, in the novel approach, "particular processes" as all processes that have the same active independent energy components are studied at "various conditions"; in other words, all conditions that same energy amount is applied to the system. One of the advantages of this novel approach is that the volume of the needed calculations will be decreased mainly in comparison with the Boltzmann entropy equation. Dependence of the entropy and rate of the energy components is gained from the novel energy conservation principle. The gained relation, expressed by energy components of the system, is considered with no constraints on the structure of the system but has a common basis with the Boltzmann entropy equation. In fact, by using a novel macroscopic-statistical approach, the entropy variation of a physical system is studied.

From Green-Kubo to the full Boltzmann kinetic approach to heat transport in crystals and glasses

Abstract: We show that vertex corrections to the quasi-harmonic Green-Kubo theory of heat transport in insulators naturally lead to a generalisation of the expression for the conductivity that could be derived from the linearized Boltzmann equation, when the effects of the full scattering matrix are accounted for. Our results, which are obtained from the Mori-Zwanzig memory-function formalism, provide a fully ab initio derivation of the linearized Boltzmann transport equation and establish a connection between two recently proposed unified approaches to heat transport in insulating crystals and glasses.

Replica symmetry breaking for Ulam's problem

Abstract: We study increasing subsequences (IS) for an ensemble of sequences given by permutation of numbers {1,2,...,n}. We consider a Boltzmann ensemble at temperature T. Thus each IS appears with the corresponding Boltzmann probability where the energy is the negative length -l of the IS. For T -> 0, only ground states, i.e. longest IS (LIS) contribute, also called Ulam's problem. We introduce an algorithm which allows us to directly sample IS in perfect equilibrium in polynomial time, for any given sequence and any temperature. Thus, we can study very large sizes. We obtain averages for the first and second moments of number of IS as function of $n$ and confirm analytical predictions. Furthermore, we analyze for low temperature $T$ the sampled ISs by computing the distribution of overlaps and performing hierarchical cluster analyses. In the thermodynamic limit the distribution of overlaps stays broad and the configuration landscape remains complex. Thus, Ulam's problem exhibits replica symmetry breaking. This means it constitutes a model with complex behavior which can be studied numerically exactly in a highly efficient way, in contrast to other RSB-showing models, like spin glasses or NP-hard optimization problems, where no fast exact algorithms are known.

More on the demons of thermodynamics

Abstract: Katie Robertson’s article gives a delightful overview of the vanquishing of demons haunting thermodynamics (Physics Today, November 2021, page 44). We want to add that Maxwell’s demon plays a special role in physics apart from concerns about vanquishing. Maxwell’s demon reveals a subtle link between information acquisition and thermodynamics.Over the past two or so decades, that link has provided inspiration for the development of a robust field, stochastic thermodynamics, which enables analysis of the energetics of nonmacroscopic systems with information feedback. Stochastic thermodynamics formalizes what the demon has taught us informally—namely, that information is a resource that can enhance the ability of a system to do work, and erasure of each bit of information in the demon’s memory increases entropy by kB ln 2 (with kB being Boltzmann’s constant), assuring the sanctity of the second law. Without Maxwell’s demon, it is questionable whether stochastic thermodynamics and a host of interesting nonmacroscopic experimental results would exist today. Section:ChooseTop of page <<© 2023 American Institute of Physics.

Application of Boltzmann kinetic equations to model X-ray-created warm dense matter and plasma

Abstract: In this review, we describe the application of Boltzmann kinetic equations for modelling warm dense matter and plasma formed after irradiation of solid materials with intense femtosecond X-ray pulses. Classical Boltzmann kinetic equations are derived from the reduced N-particle Liouville equations. They include only single-particle densities of ions and free electrons present in the sample. The first version of the Boltzmann kinetic equation solver was completed in 2006. It could model non-equilibrium evolution of X-ray-irradiated finite-size atomic systems. In 2016, the code was adapted to study plasma created from X-ray-irradiated materials. Additional extension of the code was then also performed, enabling simulations in the hard X-ray irradiation regime. In order to avoid treatment of a very high number of active atomic configurations involved in the excitation and relaxation of X-ray-irradiated materials, an approach called ‘predominant excitation and relaxation path’ (PERP) was introduced. It limited the number of active atomic configurations by following the sample evolution only along most PERPs. The performance of the Boltzmann code is illustrated in the examples of X-ray-heated solid carbon and gold. Actual model limitations and further model developments are discussed. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.

Thermodynamic and electron transport properties of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Ca</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi>Ru</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>7</mml:m

Abstract: This work demonstrates a first-principles-based approach to obtaining finite temperature thermal and electronic transport properties which can be employed to model and understand mesoscale structural evolution during electronic, magnetic, and structural phase transitions. A computationally tractable model was introduced to estimate electron relaxation time and its temperature dependence. The model is applied to ${\mathrm{Ca}}_{3}{\mathrm{Ru}}_{2}{\mathrm{O}}_{7}$ with a focus on understanding its electrical resistivity across the electronic phase transition at 48 K. A quasiharmonic phonon approach to the lattice vibrations was employed to account for thermal expansion while the Boltzmann transport theory including spin-orbit coupling was used to calculate the electron-transport properties, including the temperature dependence of electrical conductivity.

On the Emergence of Quantum Boltzmann Fluctuation Dynamics near a Bose–Einstein Condensate

Abstract: In this work, we study the quantum fluctuation dynamics in a Bose gas on a torus $$\Lambda =(L{\mathbb {T}})^3$$ that exhibits Bose–Einstein condensation, beyond the leading order Hartree–Fock–Bogoliubov (HFB) theory. Given a Bose–Einstein condensate (BEC) with density $$N\gg 1$$ surrounded by thermal fluctuations with density 1, we assume that the system dynamics is generated by a Hamiltonian with mean-field scaling. We derive a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the BEC dynamics and the HFB dynamics, with rigorous error control. Given a quasifree initial state, we determine the time evolution of the centered correlation functions $$\langle a\rangle $$ , $$\langle aa\rangle -\langle a\rangle ^2$$ , $$\langle a^+a\rangle -|\langle a\rangle |^2$$ at mesoscopic time scales $$t\sim \lambda ^{-2}$$ , where $$0<\lambda \ll 1$$ is the coupling constant determining the HFB interaction, and a, $$a^+$$ denote annihilation and creation operators. While the BEC and the HFB fluctuations both evolve at a microscopic time scale $$t\sim 1$$ , the Boltzmann dynamics is much slower, by a factor $$\lambda ^2$$ . For large but finite N, we consider both the case of fixed system size $$L\sim 1$$ , and the case $$L\sim \lambda ^{-2-}$$ . In the case $$L\sim 1$$ , we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in $${\mathbb {Q}}$$ ; this phenomenon is reminiscent of the Talbot effect. For the case $$L\sim \lambda ^{-2-}$$ , we prove that the collision operator is well approximated by the expression predicted in the literature. In either of those cases, we have $$\lambda \sim \Big (\frac{\log \log N}{\log N}\Big )^{\alpha }$$ , for different values of $$\alpha >0$$ .

Differentiable Rotamer Sampling with Molecular Force Fields

Abstract: Molecular dynamics is the primary computational method by which modern structural biology explores macromolecule structure and function. Boltzmann generators have been proposed as an alternative to molecular dynamics, by replacing the integration of molecular systems over time with the training of generative neural networks. This neural network approach to MD samples rare events at a higher rate than traditional MD, however critical gaps in the theory and computational feasibility of Boltzmann generators significantly reduce their usability. Here, we develop a mathematical foundation to overcome these barriers; we demonstrate that the Boltzmann generator approach is sufficiently rapid to replace traditional MD for complex macromolecules, such as proteins in specific applications, and we provide a comprehensive toolkit for the exploration of molecular energy landscapes with neural networks.

Lattice Boltzmann models for hydraulic engineering problems

Abstract: Lattice Boltzmann models have been gaining great attention among researchers in the hydraulic engineering field because of their simplicity and proven potential of applicability for various problems in the field. This chapter is aimed at providing a brief implication of the lattice Boltzmann models for hydraulic engineering problems. The intrinsic nature of almost all lattice Boltzmann models is very suitable for closed conduit hydraulics research, while the limited types of the lattice Boltzmann models could be used for open channel hydraulics at different scales. But the contributions of those models are invaluable for science and engineering practices. The chapter provides the validated solutions of the simple hydraulic problems representing their complex ones with the existing lattice Boltzmann models in the literature and their future outlook on the current research trends.

On the Sample Complexity of Quantum Boltzmann Machine Learning

Abstract: Quantum Boltzmann machines (QBMs) are machine-learning models for both classical and quantum data. We give an operational definition of QBM learning in terms of the difference in expectation values between the model and target, taking into account the polynomial size of the data set. By using the relative entropy as a loss function this problem can be solved without encountering barren plateaus. We prove that a solution can be obtained with stochastic gradient descent using at most a polynomial number of Gibbs states. We also prove that pre-training on a subset of the QBM parameters can only lower the sample complexity bounds. In particular, we give pre-training strategies based on mean-field, Gaussian Fermionic, and geometrically local Hamiltonians. We verify these models and our theoretical findings numerically on a quantum and a classical data set. Our results establish that QBMs are promising machine learning models trainable on future quantum devices.

Boltzmann Brain Probability – a Biological Solution for Cosmology, Physics and Philosophy

Abstract: The Boltzmann Brain concept is derived from the application of the theory of entropy to suggest that based on a possible probability of a thermal or quantum fluctuation, a spontaneous brain with human memories and cognition will come into existence before the Heath Death of the universe. This had led to cosmological and philosophical discussions regarding our understanding of the physics of universe, existentialism, and the anthropic principle. Calculations to quantify Boltzmann Brain genesis have so-far been circumscribed through the application of cosmological theories of thermal fluctuations in Minkowski Space or quantum fluctuations, typically via nucleation in de Sitter Space. Within these physics-based approaches, the assumptions of brain generation are on non-specific, non-biological relative probabilities of coming into existence though a variety of cosmological models. I apply a probabilistic model of spontaneous human brain genesis with real-world biological data to derive a quantifiable value for Boltzmann Brain genesis. This biologically derived Boltzmann Brain can be used to frame approaches to cosmological and philosophical solutions for Boltzmann brain considerations.

Plasma kinetics: Discrete Boltzmann modelling and Richtmyer-Meshkov instability

Abstract: A discrete Boltzmann model (DBM) for plasma kinetics is proposed. The DBM contains two physical functions. The first is to capture the main features aiming to investigate and the second is to present schemes for checking thermodynamic non-equilibrium (TNE) state and describing TNE effects. For the first function, mathematically, the model is composed of a discrete Boltzmann equation coupled by a magnetic induction equation. Physically, the model is equivalent to a hydrodynamic model plus a coarse-grained model for the most relevant TNE behaviors including the entropy production rate. The first function is verified by recovering hydrodynamic non-equilibrium (HNE) behaviors of a number of typical benchmark problems. Extracting and analyzing the most relevant TNE effects in Orszag-Tang problem are practical applications of the second function. As a further application, the Richtmyer-Meshkov instability with interface inverse and re-shock process is numerically studied. It is found that, in the case without magnetic field, the non-organized momentum flux shows the most pronounced effects near shock front, while the non-organized energy flux shows the most pronounced behaviors near perturbed interface. The influence of magnetic field on TNE effects shows stages: before the interface inverse, the TNE strength is enhanced by reducing the interface inverse speed; while after the interface inverse, the TNE strength is significantly reduced. Both the global averaged TNE strength and entropy production rate contributed by non-organized energy flux can be used as physical criteria to identify whether or not the magnetic field is sufficient to prevent the interface inverse.

Quantum states in materials with complex topological electronic and magnetic structure

Abstract: Η σύνθετη τοπολογική ηλεκτρονική και μαγνητική δομή των υλικών είναι ένα πεδίο έρευνας που αποκτά όλο και μεγαλύτερη σημασία τα τελευταία χρόνια, λόγω της εφαρμογής του στο πεδίο της σπιντρονικής, με πιθανές προεκτάσεις στην πληροφορία της τεχνολογίας. Στην παρούσα διδακτορική διατριβή, κύριο στόχο αποτελεί η θεωρητική και υπολογιστική μελέτη φαινομένων μεταφοράς του σπιν σε τοπολογικές δομές. Οι προσομοιώσεις μας βασίζονται σε υπολογισμούς υλικών από πρώτες αρχές εφαρμόζοντας τη θεωρία ηλεκτρονικής σκέδασης. Αρχικά, επικεντρωνόμαστε στο φαινόμενο της ροπής στρέψης σπιν σε τοπολογικούς μονωτές εμπλουτισμένους με μαγνητικές προσμίξεις. Μελετούμε τη ροπή στρέψης σπιν που ασκείται στη μαγνητική ροπή σιδηρομαγνητικά συζευγμένων προσμίξεων, και συγκεκριμένα μετάλλων μετάβασης (Cr, Mn, Fe, και Co), στην επιφάνεια του τοπολογικού μονωτή Bi2Te3, ως απόκριση σε ηλεκτρικό ρεύματος στην επιφάνεια. Οι ιδιότητες σκέδασης των επιφανειακών καταστάσεων στις μαγνητικές προσμίξεις υπολογίζονται με τη μέθοδο Korringa-Kohn-Rostoker (KKR) συναρτήσεων Green, ενώ οι υπολογισμοί της ροπής στρέψης σπιν πραγματοποιούνται συνδυάζοντας τα αποτελέσματα της KKR στην επιφάνεια Fermi και το ρυθμό σκέδασης με την ημικλασική γραμμικοποιημένη εξίσωση Boltzmann. Συζητάμε τη συσχέτιση της ροπής στρέψης σπιν με το ρεύμα σπιν, αναλύοντας τη συνεισφορά της ροής σπιν στη ροπή στρέψης σπιν στις προσμίξεις. Επιπλέον, εξετάζουμε πώς σχετίζεταιη ροπή στρέψης σπιν με την αντίσταση και την παραγωγή θερμότητας Joule. Σύμφωνα με τα αποτελέσματά μας, τα συστήματα αυτά είναι ευνοϊκά για σπιντρονικές εφαρμογές. Ειδικότερα, προβλέπουμε ότι το σύστημα Mn/Bi2Te3 είναι το πλέον υποσχόμενο μεταξύ των συστημάτων που μελετήσαμε για εφαρμογές της ροπής στρέψης σπιν. Στη συνέχεια, επικεντρωνόμαστε στη μελέτη διδιάστατων μαγνητικών σκυρμιονίων, τα οποία είναι τοπολογικά σολιτόνια σε σιδηρομαγνητικά υμένια και τα οποία συμπεριφέρονται ως σωματίδια που δύνανται να σχηματιστούν, μεταφερθούν και ανιχνευθούν. Για τη μελέτη αυτή, βασιστήκαμε στη μέθοδο KKR και πραγματοποιήσαμε υπολογισμούς θεωρίας συναρτησιακού της μη συγγραμικής πυκνότητας σπιν για το σχηματισμό ευσταθών μαγνητικώνσκυρμιονίων σε υπέρλεπτα υμένια Pd/Fe/Ir(111). Κατόπιν, επιλύοντας την αυτοσυνεπή εξίσωση Boltzmann, εξετάζουμε το τοπολογικό φαινόμενο Hall, το οποίο προκαλείται από τη σκέδαση των ηλεκτρονίων σε συστήματα σκυρμιονίων. Η μελέτη του τοπολογικού φαινομένου Hall είναι θεμελιώδους σημασίας σε τέτοιου είδους συστήματα, καθώς το φαινόμενο αυτό αποτελεί μία από τις βασικές μεθόδους για την ανίχνευση μαγνητικών σκυρμιονίων. Παρουσιάζουμε την αντίσταση και τη γωνία Hall του συστήματος, και εξετάζουμε την εξάρτηση του τοπολογικού φαινομένου Hall από το βαθμό αταξίας του δείγματος, εισάγοντάς τον στους υπολογισμούς μας μέσω ενός επιπλέον όρου ηλεκτρονικής σκέδασης. Τα ευρήματά μας προβλέπουν μία ισχυρή εξάρτηση του τοπολογικού φαινομένου Hall από τοβαθμό αταξίας του δείγματος.

Gradient Decay in the Boltzmann theory of Non-isothermal boundary

Abstract: We consider the Boltzmann equation in convex domain with non-isothermal boundary of diffuse reflection. For both unsteady/steady problems, we construct solutions belong to $W^{1,p}_x$ for any $p<3$. We prove that the unsteady solution converges to the steady solution in the same Sobolev space exponentially fast as $t \rightarrow \infty$.

A self-gravitating system composed of baryonic and dark matter analysed from the post-Newtonian Boltzmann equations

Abstract: We study the Jeans gravitational instability for a mixture of baryonic and dark matter particles, in the post-Newtonian approximation. We adopt a kinetic model consisting of a coupled system of post-Newtonian collisionless Boltzmann equations, for each species, coupled to the post-Newtonian Poisson equations. We derive the stability criterion, accounting for both post-Newtonian corrections and the presence of dark matter. It is shown that both effects give rise to smaller Jeans masses, in comparison with the standard Jeans criterion, meaning that a smaller mass is needed to begin the gravitational collapse. Taking advantage of that, we confront the model with the observational stability of Bok globules, and show that the model correctly reproduces the data.

Information scrambling of the dilute Bose gas at low temperature

Abstract: We calculate the quantum Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ and butterfly velocity ${v}_{B}$ in the dilute Bose gas at temperature $T$ deep in the Bose-Einstein condensation phase. The generalized Boltzmann equation approach is used for calculating out-of-time-ordered correlators, from which ${\ensuremath{\lambda}}_{L}$ and ${v}_{B}$ are extracted. At very low temperature where elementary excitations are phonon-like, we find ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}{T}^{5}$ and ${v}_{B}\ensuremath{\sim}c$, the sound velocity. At relatively high temperature, we have ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}T$ and ${v}_{B}\ensuremath{\sim}c{(T/{T}_{*})}^{0.23}$. We find that ${\ensuremath{\lambda}}_{L}$ is always comparable to the damping rate of a quasiparticle, whose energy depends suitably on $T$. The chaos diffusion constant ${D}_{L}={v}_{B}^{2}/{\ensuremath{\lambda}}_{L}$, on the other hand, differs from the energy diffusion constant ${D}_{E}$. We find ${D}_{E}\ensuremath{\ll}{D}_{L}$ at very low temperature and ${D}_{E}\ensuremath{\gg}{D}_{L}$ otherwise.

Tri-Resonant Leptogenesis

Abstract: We present a class of leptogenesis models where the light neutrinos acquire their observed mass through a symmetry-motivated construction. We consider an extension of the Standard Model, which includes three singlet neutrinos which have mass splittings comparable to their decay widths. We show that this tri-resonant structure leads to an appreciable increase in the observed CP asymmetry over that found previously in typical bi-resonant models. To analyse such tri-resonant scenarios, we solve a set of coupled Boltzmann equations, crucially preserving the variations in the relativistic degrees of freedom. We highlight the fact that small variations at high temperatures can have major implications for the evolution of the baryon asymmetry when the singlet neutrino mass scale is below $100$ GeV. We then illustrate how this variation can significantly affect the ability to find successful leptogenesis at these low masses. Finally, the parameter space for viable leptogenesis is delineated, and comparisons are made with current and future experiments.