Entropy Sn Research Articles

Temporal Evolution of Morphology, Composition, and Structure in the Formation of Colloidal High-Entropy Intermetallic Nanoparticles.

Morphology-controlled nanoparticles of high entropy intermetallic compounds are quickly becoming high-value targets for catalysis. Their ordered structures with multiple distinct crystallographic sites, coupled with the "cocktail effect" that emerges from randomly mixing a large number of elements, yield catalytic active sites capable of achieving advanced catalytic functions. Despite this growing interest, little is known about the pathways by which high entropy intermetallic nanoparticles form and grow in solution. As a result, controlling their morphology remains challenging. Here, we use the high entropy intermetallic compound (Pd,Rh,Ir,Pt)Sn, which adopts a NiAs-related crystal structure, as a model system for understanding how nanoparticle morphology, composition, and structure evolve during synthesis in solution using a slow-injection reaction. By performing a time-point study, we establish the initial formation of palladium-rich cube-like Pd-Sn seeds onto which the other metals deposit over time, concomitant with continued incorporation of tin. For (Pd,Rh,Ir,Pt)Sn, growth occurs on the corners, resulting in a sample having a mixture of flower-like and cube-like morphologies. We then synthesize and characterize a library of 14 distinct intermetallic nanoparticle systems that comprise all possible binary, ternary, and quaternary constituents of (Pd,Rh,Ir,Pt)Sn. From these studies, we correlated compositions, morphologies, and growth pathways with the constituent elements and their competitive reactivities, ultimately mapping out a framework that rationalizes the key features of the high entropy (Pd,Rh,Ir,Pt)Sn intermetallic nanoparticles based on those of their simpler constituents. We then validated these design guidelines by applying them to the synthesis of a morphologically pure variant of flowerlike (Pd,Rh,Ir,Pt)Sn particles as well as a series of (Pd,Rh,Ir,Pt)Sn particles with tunable morphologies based on control of composition.

Investigation of the Interfacial Reactions between the CoCuFeNi High Entropy Alloy and Sn Solder

The CoCuFeNi high entropy alloy (HEA) has excellent electrical, thermal, and mechanical properties. In electronic packaging technology, Sn is the major element of lead-free solders. In this study, we used the CoCuFeNi HEA as the substrate and Sn as the solder and investigated the liquid/solid interfacial reactions of the Sn/CoCuFeNi system at 300, 375, and 450 °C for 30, 60, 100, 150, 360, and 480 min. The results indicated that the (Fe, Co)Sn2 phase was formed in the Sn/CoCuFeNi couples for all various reaction temperatures and at different durations. Additionally, the (Co, Ni)Sn2 phase was precipitated at the solder side and near the (Fe, Co)Sn2 phase when the reaction time increased. The thickness of the (Fe, Co)Sn2 phase increased with the increase in reaction temperature and time, and it was proportional to the square root of the reaction time. Overall, our results showed that the growth mechanism of the (Fe, Co)Sn2 phase was diffusion-controlled in the Sn/CoCuFeNi couples.

A novel subgenotype I3 of hepatitis B virus in Guangxi, China: a 15-year follow-up study.

Genotype I of hepatitis B virus (HBV) was proposed recently following sequencing of complete HBV genomes from Vietnam and Laos. However, its long-term molecular evolution is unknown. The objectives of this study were to study the molecular evolution of this genotype from an asymptomatic HBsAg carrier from the Long An cohort over a 15-year period was studied using both NGS and clone-based sequencing. The number of complete genome sequences obtained in 2004, 2007, 2013, and 2019 are 17, 20, 19, and 10, respectively. All strains belong to subgenotype I1, except for six (five from 2007 and one from 2019) and 8 further strains from 2007 which form a cluster branching out from other subgenotype I sequences, supported by a 100% bootstrap value. Based on complete genome sequences, all of the estimated intragroup nucleotide divergence values between these strains and HBV subgenotypes I1-I2 exceed 4%. These strains are recombinants between genotype I1 and subgenotype C but the breakpoints vary. The median intrahost viral evolutionary rate in this carrier was 3.88E-4 substitutions per site per year. The Shannon entropy (Sn) ranged from 0.55 to 0.88 and the genetic diversity, D, ranged from 0.0022 to 0.0041. In conclusion, our data provide evidence of novel subgenotypes. Considering that the 8 strains disappeared after 2007, while one of the 6 strains appears again in 2019, we propose these 6 strains as a new subgenotype, provisionally designated HBV subgenotype I3 and the 8 strains as aberrant genotype.

Analysis of entire hepatitis B virus genomes reveals reversion of mutations to wild type in natural infection, a 15year follow-up study.

It has been reported that some mutations in the genome of hepatitis B virus (HBV) may predict the outcome of the virus infection. However, evolutionary data derived from long-term longitudinal analysis of entire HBV genomes using next generation sequencing (NGS) remain rare. In this study, serum samples were collected from asymptomatic hepatitis B surface antigen (HBsAg) carriers from a long-term prospective cohort. The entire HBV genome was amplified by polymerase chain reaction (PCR) and sequenced using NGS. Twenty-eight time series serum samples from nine subjects were successfully analysed. The Shannon entropy (Sn) ranged from 0 to 0.89, with a median value of 0.76, and the genetic diversity (D) ranged from 0 to 0.013, with a median value of 0.004. Intrahost HBV viral evolutionary rates ranged from 2.39E-04 to 3.11E-03. Double mutations at nt1762(A→T) and 1764(G→A) and a stop mutation at nt1896(G→A) were seen in all sequences from subject BO129 in 2007. However, in 2019, most sequences were wild type at these positions. Deletions between nt 2920-3040 were seen in all sequences from subject TS115 in 2007 and 2013 but these were not present in 2004 or 2019. Some sequences from subject CC246 had predicted escape substitutions (T123N, G145R) in the surface protein in 2004, 2013 and 2019 but none of the sequences from 2007 had these changes. In conclusion, HBV mutations may revert to wild type in natural infection. Clinicians should be wary of predicting long-term prognoses on the basis of the presence of mutations.

Time-independence of gravitational R\xe9nyi entropies and unitarity in quantum gravity

The Hubeny-Rangamani-Takayanagi surface γHRT computing the entropy S(D) of a domain of dependence D on an asymptotically AdS boundary is known to be causally inaccessible from D. We generalize this gravitational result to higher replica numbers n > 1 by considering the replica-invariant surfaces (aka ‘splitting surfaces’) γ of real-time replica-wormhole saddle-points computing Rényi entropies Sn(D) and showing that there is a sense in which D must again be causally inaccessible from γ when the saddle preserves both replica and conjugation symmetry. This property turns out to imply the Sn(D) to be independent of any choice of any Cauchy surface ΣD for D, and also that the Sn(D) are independent of the choice of boundary sources within D. This is a key hallmark of unitary evolution in any dual field theory. Furthermore, from the bulk point of view it adds to the evidence that time evolution of asymptotic observables in quantum gravity is implemented by a unitary operator in each baby universe superselection sector. Though we focus here on pure Einstein-Hilbert gravity and its Kaluza-Klein reductions, we expect the argument to extend to any two-derivative theory who satisfies the null convergence condition. We consider both classical saddles and the effect of back-reaction from quantum corrections.

R\xe9nyi entropies and area operator from gravity with Hayward term

In the context of the holographic duality, the entanglement entropy of ordinary QFT in a subregion in the boundary is given by a quarter of the area of an minimal surface embedded in the bulk spacetime. This rule has been also extended to a suitable one-parameter generalization of the von-Neuman entropy Ŝn that is related to the Rényi entropies Sn, as given by the area of a cosmic brane minimally coupled with gravity, with a tension related to n that vanishes as n → 1, and moreover, this parameter can be analytically extended to arbitrary real values. However, the brane action plays no role in the duality and cannot be considered a part of the theory of gravity, thus it is used as an auxiliary tool to find the correct background geometry.In this work we study the construction of the gravitational (reduced) density matrix from holographic states, whose wave-functionals are described as euclidean path integrals with arbitrary conditions on the asymptotic boundaries, and argue that in general, a non-trivial Hayward term must be haven into account. So we propose that the gravity model with a coupled Nambu-Goto action is not an artificial tool to account for the Rényi entropies, but it is present in the own gravity action through a Hayward term. As a result we show that the computations using replicas simplify considerably and we recover the holographic prescriptions for the measures of entanglement entropy; in particular, derive an area law for the original Rényi entropies (Sn) related to a minimal surface in the n replicated spacetime. Moreover, we show that the gravitational modular flow contains the area operator and can explain the Jafferis-Lewkowycz-Maldacena-Suh proposal.

Information Entropy and Information Distances in Atoms, Nuclei and Bosonic Systems

The universal property for the information entropy S = a + h In Ζ is verified for atoms using a systematic study with Roothaan-Hartree-Fock (RHF) wave functions with atomic number Ζ — 2 — 54. The above relation was proposed previously for atoms, nuclei, atomic clusters and correlated atoms in a trap. Kullback-Leibler relative entropy Κ and Jensen-Shannon divergence J are employed to compare RHF with Thomas-Fermi (TF) density of atoms as well as another phenomenological density obtained by Sagar et al. Two-body density distributions in position- and momentum-space are used to calculate and compare the corresponding information entropies for correlated and uncorrelated nuclei and bosonic systems (correlated atoms in a trap). It is seen that short-range correlations (SRC) increase the values of S. One-body information entropy entropy S\ is compared with two-body information entropy and a conjecture is made for TV-body information entropy SN- The entropy Κ and the divergence J are also used to evaluate the information distance between correlated and uncorrelated densities both at the one- and the two-body levels for nuclei and trapped Bose gases.

Flat entanglement spectra in fixed-area states of quantum gravity

We use the Einstein-Hilbert gravitational path integral to investigate gravita- tional entanglement at leading order O(1/G). We argue that semiclassical states prepared by a Euclidean path integral have the property that projecting them onto a subspace in which the Ryu-Takayanagi or Hubeny-Rangamani-Takayanagi surface has definite area gives a state with a flat entanglement spectrum at this order in gravitational perturbation theory. This means that the reduced density matrix can be approximated as proportional to the identity to the extent that its Renyi entropies Sn are independent of n at this order. The n-dependence of Sn in more general states then arises from sums over the RT/HRT- area, which are generally dominated by different values of this area for each n. This provides a simple picture of gravitational entanglement, bolsters the connection between holographic systems and tensor network models, clarifies the bulk interpretation of alge- braic centers which arise in the quantum error-correcting description of holography, and strengthens the connection between bulk and boundary modular Hamiltonians described by Jafferis, Lewkowycz, Maldacena, and Suh.

Holographic Renyi entropy from quantum error correction

We study Renyi entropies Sn in quantum error correcting codes and compare the answer to the cosmic brane prescription for computing {tilde{S}}_nequiv {n}^2{partial}_nleft(frac{n-1}{n}{S}_nright) . We find that general operator algebra codes have a similar, more general prescription. Notably, for the AdS/CFT code to match the specific cosmic brane prescription, the code must have maximal entanglement within eigenspaces of the area operator. This gives us an improved definition of the area operator, and establishes a stronger connection between the Ryu-Takayanagi area term and the edge modes in lattice gauge theory. We also propose a new interpretation of existing holographic tensor networks as area eigenstates instead of smooth geometries. This interpretation would explain why tensor networks have historically had trouble modeling the Renyi entropy spectrum of holographic CFTs, and it suggests a method to construct holographic networks with the correct spectrum.

Higher baseline viral diversity correlates with lower HBsAg decline following PEGylated interferon-alpha therapy in patients with HBeAg-positive chronic hepatitis B.

BackgroundViral diversity seems to predict treatment outcomes in certain viral infections. The aim of this study was to evaluate the association between baseline intra-patient viral diversity and hepatitis B surface antigen (HBsAg) decline following PEGylated interferon-alpha (Peg-IFN-α) therapy.Patients and methodsTwenty-six HBeAg-positive patients who were treated with Peg-IFN-α were enrolled. Nested polymerase chain reaction (PCR), cloning, and sequencing of the hepatitis B virus S gene were performed on baseline samples, and normalized Shannon entropy (Sn) was calculated as a measure of small hepatitis B surface protein (SHBs) diversity. Multiple regression analysis was used to estimate the association between baseline Sn and HBsAg decline.ResultsOf the 26 patients enrolled in the study, 65.4% were male and 61.5% were infected with hepatitis B virus genotype B. The median HBsAg level at baseline was 4.5 log10 IU/mL (interquartile range: 4.1–4.9) and declined to 3.0 log10 IU/mL (interquartile range: 1.7–3.9) after 48 weeks of Peg-IFN-α treatment. In models adjusted for baseline alanine aminotransferase (ALT) and HBsAg, the adjusted coefficients (95% CI) for ΔHBsAg and relative percentage HBsAg decrease were −1.3 (−2.5, −0.2) log10 IU/mL for higher SHBs diversity (Sn≥0.58) patients and −26.4% (−50.2%, −2.5%) for lower diversity (Sn<0.58) patients. Further analysis showed that the “a” determinant upstream flanking region and the first loop of the “a” determinant (nucleotides 341–359, 371–389, and 381–399) were the main sources of higher SHBs diversity.ConclusionBaseline intra-patient SHBs diversity was inverse to HBsAg decline in HBeAg-positive chronic hepatitis B (CHB) patients receiving Peg-IFN-α monotherapy. Also, more sequence variations within the “a” determinant upstream flanking region and the first loop of the “a” determinant were the main sources of the higher SHBs diversity.

Discrete gravity on random tensor network and holographic R\xe9nyi entropy

In this paper we apply the discrete gravity and Regge calculus to tensor networks and Anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We construct the boundary many-body quantum state |Ψ〉 using random tensor networks as the holographic mapping, applied to the Wheeler-deWitt wave function of bulk Euclidean discrete gravity in 3 dimensions. The entanglement Rényi entropy of |Ψ〉 is shown to holographically relate to the on-shell action of Einstein gravity on a branch cover bulk manifold. The resulting Rényi entropy Sn of |Ψ〉 approximates with high precision the Rényi entropy of ground state in 2-dimensional conformal field theory (CFT). In particular it reproduces the correct n dependence. Our results develop the framework of realizing the AdS3/CFT2 correspondence on random tensor networks, and provide a new proposal to approximate the CFT ground state.

Relative entanglement entropies in 1 + 1-dimensional conformal field theories

We study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy S(ρ1∥ρ0) between two given reduced density matrices ρ1 and ρ0 of a quantum field theory, we employ the replica trick which relies on the path integral representation of Tr(ρ1ρ0n − 1) and define a set of Rényi relative entropies Sn(ρ1∥ρ0). We compute these quantities for integer values of the parameter n and derive via the replica limit the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator i∂ϕ, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement.

Contribution of next generation sequencing to early detection of cytomegalovirus UL97 emerging mutants and viral subpopulations analysis in kidney transplant recipients

BackgroundCytomegalovirus (CMV) is the major opportunistic virus encountered after transplantation, and resistant variants challenge antiviral treatment. We studied the emergence and evolution of the canonical UL97 L595S mutation in four kidney recipients by comparing Sanger sequencing with a specific next-generation sequencing (NGS) assay, and assessed the global evolution of UL97 gene variability. Study designPlasmids harbouring wild-type and/or L595S mutated UL97 genes were used to assess the analytical performances of NGS assay. UL97 gene was retrospectively analysed in patients’ samples drawn during CMV infection follow-up, Shannon entropy (Sn) was calculated and phylogenetic analyses were performed. ResultsWild-type and L595S plasmids PCR products were mixed to obtain L595S concentrations of 0, 1, 2, 5, 10, 20 and 100%. Mean triplicate NGS results were 0, 0.71, 1.79, 5.30, 13.17, 17 and 100%, respectively, while Sanger sequencing only detected L595S when above 20%. The NGS mean error rate was 0.196±0.07%. In the four patients, emergence of L595S mutation under ganciclovir treatment was followed-up. After foscarnet rescue therapy, leading to undetectable CMV viral load, in two patients, L595S mutant re-emerged, but was only detected by NGS technology (14% and 9.6%). Using NGS data, phylogenetic trees and Sn showed a complex evolution of concomitant viral subpopulations. ConclusionsNGS technology allowed a deeper discrimination of the emergence and persistence of a drug resistance mutation, which could be pertinent to investigate when routine Sanger sequencing detects only wild-type strains. Moreover, NGS improved sensitivity helps in studying viral abundance, dynamics and diversity, less approachable with Sanger sequencing.

Chaos and Case or Implementation of the Program of Evolution of Space

The classical concept of entropy is shown not to reflect the essence of the second law of thermodynamics both phenomenologically and theoretically. A new concept of entropy Sn, is given, which offers a particular physical meaning of entropy and provides an experimental determination of the entropy value. At the same time, it is suggested not to define entropy but irreversibility——as the key characteristic of thermodynamics second law for real processes. In addition, this article demonstrates that it is wrong to mechanically transfer the statistical concept and mathematical formula of the entropy to the field of real systems. This approach has led to incorrect basis for a number of theoretical sciences, which study both the phenomenon of life and the different systems—automates, computers, robots etc.—which model life’s separate functions. This allows one to avoid many attempts at defining the phenomenon of life via entropy characteristics and give definitions of life and an elementary structure of both life itself and devices that simulate life (automata, computers, and robots). This, in turn, allows one to assign a substantive sense to the mathematical character of corresponding sciences (information theory, cybernetics, system theory, etc.) and define operational a single basic concepts of information, knowledge, meaning, and control, as well as to formulate operational concepts of order, organization, and self-organization and implement a new approach to problems of biology, biophysics, information theory, cybernetics, and system theory. Additionally, on the basis of scientific facts the evolution of the universe is considered as an implementation of the existing program, which allows a new approach to problems of world outlook and science, on the one hand, and a more efficient approach to the problem of genome decoding and to analysis of brain structure and its functioning in the immediate scientific practice, on the other hand.

Entropy is a consequence of a discrete time

While the basic microscopic physical laws are time reversible, the arrow of time and time irreversibility appears only at the macroscopic physical laws by the second law of thermodynamics with its entropy term S. It is the attempt of the present work to bridge the microscopic physical world with its macroscopic one with an alternative approach than the statistical mechanics theory of Gibbs and Boltzmann. For simplicity a “classical”, single particle in a one dimensional space is selected. In addition, it is assumed that time is discrete with constant step size. As a consequence time irreversibility at the microscopic level is obtained if the present force is of complex nature (F(r) ≠ const). In order to compare this discrete time irreversible mechanics with its classical Newton analog, time reversibility is reintroduced by scaling the time steps for any given time step n by the variable sn leading to the Nosé-Hoover Lagrangian comprising a term NdfkB T In sn (kB the Boltzmann constant, T the temperature, and Ndf the number of degrees of freedom) which is defined as the microscopic entropy Sn at time point n multiplied by T. Upon ensemble averaging of the microscopic entropy in a many particles system in thermodynamic equilibrium it approximates its macroscopic counterpart known from statistical mechanics. The presented derivation with the resulting analogy between the ensemble averaged microscopic entropy and its statistical mechanics analog suggests that the entropy term itself has its root not in statistical mechanics but rather in the discreteness of time.

Constructing Entanglers in 2-Players–N-Strategies Quantum Game

In quantum games based on 2-player-N-strategies classical games, each player has a quNit (a normalized vector in an N-dimensional Hilbert space HN) upon which he applies his strategy (a matrix U∈SU(N)). The players draw their payoffs from a state . Here and J (both determined by the game’s referee) are respectively an unentangled 2-quNit (pure) state and a unitary operator such that is partially entangled. The existence of pure strategy Nash equilibrium in the quantum game is intimately related to the degree of entanglement of . Hence, it is practical to design the entangler J= J(β) to be dependent on a single real parameter β that controls the degree of entanglement of , such that its von-Neumann entropy SN(β) is continuous and obtains any value in . Designing J(β) for N=2 is quite standard. Extension to N>2 is not obvious, and here we suggest an algorithm to achieve it. Such construction provides a special quantum gate that should be a useful tool not only in quantum games but, more generally, as a special gate in manipulating quantum information protocols.

A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time

The basic microsopic physical laws are time reversible. In contrast, the second law of thermodynamics, which is a macroscopic physical representation of the world, is able to describe irreversible processes in an isolated system through the change of entropy ΔS &gt; 0. It is the attempt of the present manuscript to bridge the microscopic physical world with its macrosocpic one with an alternative approach than the statistical mechanics theory of Gibbs and Boltzmann. It is proposed that time is discrete with constant step size. Its consequence is the presence of time irreversibility at the microscopic level if the present force is of complex nature (F(r) ≠ const). In order to compare this discrete time irreversible mechamics (for simplicity a “classical”, single particle in a one dimensional space is selected) with its classical Newton analog, time reversibility is reintroduced by scaling the time steps for any given time step n by the variable sn leading to the Nosé-Hoover Lagrangian. The corresponding Nos´e-Hoover Hamiltonian comprises a term Ndf kB T ln sn (kB the Boltzmann constant, T the temperature, and Ndf the number of degrees of freedom) which is defined as the microscopic entropy Sn at time point n multiplied by T. Upon ensemble averaging this microscopic entropy Sn in equilibrium for a system which does not have fast changing forces approximates its macroscopic counterpart known from thermodynamics. The presented derivation with the resulting analogy between the ensemble averaged microscopic entropy and its thermodynamic analog suggests that the original description of the entropy by Boltzmann and Gibbs is just an ensemble averaging of the time scaling variable sn which is in equilibrium close to 1, but that the entropy

Ultra-Deep Pyrosequencing (UDPS) Data Treatment to Study Amplicon HCV Minor Variants

We have investigated the reliability and reproducibility of HCV viral quasispecies quantification by ultra-deep pyrosequencing (UDPS) methods. Our study has been divided in two parts. First of all, by UDPS sequencing of clone mixes samples we have established the global noise level of UDPS and fine tuned a data treatment workflow previously optimized for HBV sequence analysis. Secondly, we have studied the reproducibility of the methodology by comparing 5 amplicons from two patient samples on three massive sequencing platforms (FLX+, FLX and Junior) after applying the error filters developed from the clonal/control study. After noise filtering the UDPS results, the three replicates showed the same 12 polymorphic sites above 0.7%, with a mean CV of 4.86%. Two polymorphic sites below 0.6% were identified by two replicates and one replicate respectively. A total of 25, 23 and 26 haplotypes were detected by GS-Junior, GS-FLX and GS-FLX+. The observed CVs for the normalized Shannon entropy (Sn), the mutation frequency (Mf), and the nucleotidic diversity (Pi) were 1.46%, 3.96% and 3.78%. The mean absolute difference in the two patients (5 amplicons each), in the GS-FLX and GS-FLX+, were 1.46%, 3.96% and 3.78% for Sn, Mf and Pi. No false polymorphic site was observed above 0.5%.Our results indicate that UDPS is an optimal alternative to molecular cloning for quantitative study of HCV viral quasispecies populations, both in complexity and composition. We propose an UDPS data treatment workflow for amplicons from the RNA viral quasispecies which, at a sequencing depth of at least 10,000 reads per strand, enables to obtain sequences and frequencies of consensus haplotypes above 0.5% abundance with no erroneous mutations, with high confidence, resistant mutants as minor variants at the level of 1%, with high confidence that variants are not missed, and highly confident measures of quasispecies complexity.

Entanglement property in matrix product spin systems

We study the entanglement property in matrix product spin-ring systems systemically by von Neumann entropy. We find that: (i) the Hilbert space dimension of one spin determines the upper limit of the maximal value of the entanglement entropy of one spin, while for multiparticle entanglement entropy, the upper limit of the maximal value depends on the dimension of the representation matrices. Based on the theory, we can realize the maximum of the entanglement entropy of any spin block by choosing the appropriate control parameter values. (ii) When the entanglement entropy of one spin takes its maximal value, the entanglement entropy of an asymptotically large spin block, i.e. the renormalization group fixed point, is not likely to take its maximal value, and so only the entanglement entropy Sn of a spin block that varies with size n can fully characterize the spin-ring entanglement feature. Finally, we give the entanglement dynamics, i.e. the Hamiltonian of the matrix product system.

Work and heat fluctuations in two-state systems: a trajectory thermodynamicsformalism

Two-state models provide phenomenological descriptions of many differentsystems, ranging from physics to chemistry and biology. We investigate workfluctuations in an ensemble of two-state systems driven out of equilibrium underthe action of an external perturbation. We calculate the probability densityPN(W) that work equal toW is exerted uponthe system (of size N) along a given non-equilibrium trajectory and introduce a trajectorythermodynamics formalism to quantify work fluctuations in thelarge-N limit. We thendefine a trajectory entropy SN(W) that counts the number of non-equilibrium trajectoriesPN(W) = exp(SN(W)/kBT) with workequal to W and characterizes fluctuations of work trajectories around the most probable valueWmp. A trajectory free energy can also be defined, which has a minimum atW = W†, this being the value of the work that has to be efficiently sampled to quantitatively testthe Jarzynski equality. Within this formalism a Lagrange multiplier is also introduced, theinverse of which plays the role of a trajectory temperature. Our general solution forPN(W) exactly satisfies the fluctuation theorem by Crooks and allows us to investigate heatfluctuations for a protocol that is invariant under time reversal. The heat distributionis then characterized by a Gaussian component (describing small and frequentheat exchange events) and exponential tails (describing the statistics of largedeviations and rare events). For the latter, the width of the exponential tails isrelated to the aforementioned trajectory temperature. Finite-size effects to thelarge-N theory and the recovery of work distributions for finiteN are also discussed. Finally, we pay particular attention to the case ofmagnetic nanoparticle systems under the action of a magnetic fieldH where work and heat fluctuations are predicted to be observable in ramping experiments inmicro-SQUIDs.