Pub Date : 2010-03-01DOI: 10.1080/00411450.2010.533741
Wilfried Kirschenmann, L. Plagne, A. Ponçot, S. Vialle
This paper presents two parallel Simplified PN (SPN) solver implementations for both multi-core Central Processing Units (CPU) and Graphics Processing Units (GPU). For a nuclear operator such as Électricité de France (EDF), the time required to carry out nuclear reactor core simulations is rather critical when dealing with production constraints. The SPN method provides a convenient trade-off between accuracy and numerical complexity and is used in several industrial simulations. The parallelization of the SPN algorithm reduces its computation time. To solve the problem on distributed memory machines such as PC clusters, Domain Decomposition Methods have been investigated. Complementary to this approach, this work aims to use emerging massively parallel processors such as the GPUs as well as current multi-core CPUs. Based on a fine grained parallelism, this solution achieves good performances on desktop machines. Our multi-core CPU and many-core GPU implementations allow us to solve 3D SPN problems, respectively, 10 and 36 times faster than our sequential CPU reference.
本文提出了两种适用于多核中央处理器(CPU)和图形处理器(GPU)的并行简化PN (SPN)求解器实现。对于像Électricité de France (EDF)这样的核运营商来说，在处理生产限制时，进行核反应堆堆芯模拟所需的时间是相当关键的。SPN方法在精度和数值复杂性之间提供了一个方便的权衡，并在一些工业模拟中使用。SPN算法的并行化减少了算法的计算时间。为了解决PC集群等分布式存储机器上的问题，研究了领域分解方法。作为这种方法的补充，这项工作旨在使用新兴的大规模并行处理器，如gpu以及当前的多核cpu。基于细粒度并行性，该解决方案在桌面机器上实现了良好的性能。我们的多核CPU和多核GPU实现使我们能够解决3D SPN问题，分别比我们的顺序CPU参考快10倍和36倍。
{"title":"Parallel SPN on Multi-Core CPUS and Many-Core GPUS","authors":"Wilfried Kirschenmann, L. Plagne, A. Ponçot, S. Vialle","doi":"10.1080/00411450.2010.533741","DOIUrl":"https://doi.org/10.1080/00411450.2010.533741","url":null,"abstract":"This paper presents two parallel Simplified PN (SPN) solver implementations for both multi-core Central Processing Units (CPU) and Graphics Processing Units (GPU). For a nuclear operator such as Électricité de France (EDF), the time required to carry out nuclear reactor core simulations is rather critical when dealing with production constraints. The SPN method provides a convenient trade-off between accuracy and numerical complexity and is used in several industrial simulations. The parallelization of the SPN algorithm reduces its computation time. To solve the problem on distributed memory machines such as PC clusters, Domain Decomposition Methods have been investigated. Complementary to this approach, this work aims to use emerging massively parallel processors such as the GPUs as well as current multi-core CPUs. Based on a fine grained parallelism, this solution achieves good performances on desktop machines. Our multi-core CPU and many-core GPU implementations allow us to solve 3D SPN problems, respectively, 10 and 36 times faster than our sequential CPU reference.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"255 - 281"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2010.533741","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-02-04DOI: 10.1080/00411450.2011.566484
G. Hagstrom, P. Morrison
The notions of spectral stability and the spectrum for the Vlasov-Poisson system linearized about homogeneous equilibria, f 0(v), are reviewed. Structural stability is reviewed and applied to perturbations of the linearized Vlasov operator through perturbations of f 0. We prove that for each f 0 there is an arbitrarily small δf′0 in such that f 0+δf 0 is unstable. When f 0 is perturbed by an area preserving rearrangement, f 0 will always be stable if the continuous spectrum is only of positive signature, where the signature of the continuous spectrum is defined as in Morrison and Pfirsch (1992) and Morrison (2000). If there is a signature change, then there is a rearrangement of f 0 that is unstable and arbitrarily close to f 0 with f′0 in W.1,1 This result is analogous to Krein's theorem for the continuous spectrum. We prove that if a discrete mode embedded in the continuous spectrum is surrounded by the opposite signature there is an infinitesimal perturbation in Cn norm that makes f 0 unstable. If f 0 is stable we prove that the signature of every discrete mode is the opposite of the continuum surrounding it.
评述了谱稳定性的概念和Vlasov-Poisson系统关于齐次平衡f0 (v)线性化的谱。回顾了结构稳定性，并将其应用于线性化Vlasov算子的摄动。我们证明了对于每一个f 0有一个任意小的δf ' 0使得f 0+ f 0是不稳定的。当f 0受到保面积重排的扰动时，如果连续谱只有正签名，则f 0总是稳定的，其中连续谱的签名定义为Morrison and Pfirsch(1992)和Morrison(2000)。如果有一个特征变化，那么f 0的重排是不稳定的，并且在w .1,1中f ' 0任意接近于f 0。这个结果类似于连续谱的Krein定理。我们证明了如果一个嵌入在连续谱中的离散模式被相反的特征包围，那么在Cn范数中存在一个使f0不稳定的无穷小扰动。如果f 0是稳定的，我们证明了每个离散模态的特征与它周围的连续谱相反。
{"title":"On Krein-Like Theorems for Noncanonical Hamiltonian Systems with Continuous Spectra: Application to Vlasov-Poisson","authors":"G. Hagstrom, P. Morrison","doi":"10.1080/00411450.2011.566484","DOIUrl":"https://doi.org/10.1080/00411450.2011.566484","url":null,"abstract":"The notions of spectral stability and the spectrum for the Vlasov-Poisson system linearized about homogeneous equilibria, f 0(v), are reviewed. Structural stability is reviewed and applied to perturbations of the linearized Vlasov operator through perturbations of f 0. We prove that for each f 0 there is an arbitrarily small δf′0 in such that f 0+δf 0 is unstable. When f 0 is perturbed by an area preserving rearrangement, f 0 will always be stable if the continuous spectrum is only of positive signature, where the signature of the continuous spectrum is defined as in Morrison and Pfirsch (1992) and Morrison (2000). If there is a signature change, then there is a rearrangement of f 0 that is unstable and arbitrarily close to f 0 with f′0 in W.1,1 This result is analogous to Krein's theorem for the continuous spectrum. We prove that if a discrete mode embedded in the continuous spectrum is surrounded by the opposite signature there is an infinitesimal perturbation in Cn norm that makes f 0 unstable. If f 0 is stable we prove that the signature of every discrete mode is the opposite of the continuum surrounding it.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"466 - 501"},"PeriodicalIF":0.0,"publicationDate":"2010-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2011.566484","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-02-02DOI: 10.1080/00411450.2011.566502
M. Marklund, J. Zamanian, G. Brodin
The concept of phase space distribution functions and their evolution is used in the case of en enlarged phase space. In particular, we include the intrinsic spin of particles and present a quantum kinetic evolution equation for a scalar quasi-distribution function. In contrast to the proper Wigner transformation technique, for which we expect the corresponding quasi-distribution function to be a complex matrix, we introduce a spin projection operator for the density matrix in order to obtain the aforementioned scalar quasi-distribution function. There is a close correspondence between this projection operator and the Husimi (or Q) function used extensively in quantum optics. Such a function is based on a Gaussian smearing of a Wigner function, giving a positive definite distribution function. Thus, our approach gives a Wigner-Husimi quasi-distribution function in extended phase space, for which the reduced distribution function on the Bloch sphere is strictly positive. We also discuss the gauge issue and the fluid moment hierarchy based on such a quantum kinetic theory.
{"title":"Spin Kinetic Theory—Quantum Kinetic Theory in Extended Phase Space","authors":"M. Marklund, J. Zamanian, G. Brodin","doi":"10.1080/00411450.2011.566502","DOIUrl":"https://doi.org/10.1080/00411450.2011.566502","url":null,"abstract":"The concept of phase space distribution functions and their evolution is used in the case of en enlarged phase space. In particular, we include the intrinsic spin of particles and present a quantum kinetic evolution equation for a scalar quasi-distribution function. In contrast to the proper Wigner transformation technique, for which we expect the corresponding quasi-distribution function to be a complex matrix, we introduce a spin projection operator for the density matrix in order to obtain the aforementioned scalar quasi-distribution function. There is a close correspondence between this projection operator and the Husimi (or Q) function used extensively in quantum optics. Such a function is based on a Gaussian smearing of a Wigner function, giving a positive definite distribution function. Thus, our approach gives a Wigner-Husimi quasi-distribution function in extended phase space, for which the reduced distribution function on the Bloch sphere is strictly positive. We also discuss the gauge issue and the fluid moment hierarchy based on such a quantum kinetic theory.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"502 - 523"},"PeriodicalIF":0.0,"publicationDate":"2010-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2011.566502","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-01-13DOI: 10.1080/00411450.2010.529979
Z. Zainuddin, Mohammed Mahmod Shuaib
The Social Force Model is one of the most successful microscopic pedestrian models that represent the well-organized phenomena of the pedestrian flow. The model has been modified for evacuation process by incorporating physical forces when contact exists, on one hand, and incorporating factors into the preferred velocity to govern the individual's behavior corresponding to the situation under consideration (normal or evacuation) on the other hand. The latter incorporation has enhanced the ability of the model to represent the decision-making process of pedestrians. However, the variety of pedestrian's abilities to make decisions in emergency situations has not been incorporated properly into the model. In this article we enhance the decision-making capability of the independent pedestrians first by improving the assessment process of selecting an exit from the set of exits available in the physical environment by considering a new factor (crowd at exits); and second, by incorporating following crowds as a new feature for those who are independent. A simulation of an emergency situation inside a room is performed to validate our work.
{"title":"Modification of the Decision-Making Capability in the Social Force Model for the Evacuation Process","authors":"Z. Zainuddin, Mohammed Mahmod Shuaib","doi":"10.1080/00411450.2010.529979","DOIUrl":"https://doi.org/10.1080/00411450.2010.529979","url":null,"abstract":"The Social Force Model is one of the most successful microscopic pedestrian models that represent the well-organized phenomena of the pedestrian flow. The model has been modified for evacuation process by incorporating physical forces when contact exists, on one hand, and incorporating factors into the preferred velocity to govern the individual's behavior corresponding to the situation under consideration (normal or evacuation) on the other hand. The latter incorporation has enhanced the ability of the model to represent the decision-making process of pedestrians. However, the variety of pedestrian's abilities to make decisions in emergency situations has not been incorporated properly into the model. In this article we enhance the decision-making capability of the independent pedestrians first by improving the assessment process of selecting an exit from the set of exits available in the physical environment by considering a new factor (crowd at exits); and second, by incorporating following crowds as a new feature for those who are independent. A simulation of an emergency situation inside a room is performed to validate our work.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"125 1","pages":"47 - 70"},"PeriodicalIF":0.0,"publicationDate":"2010-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2010.529979","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-01-13DOI: 10.1080/00411450.2010.529630
J. Cartier, M. Peybernes
We introduce a new variational formulation of the transport equation based on a mixed form of the transport equation. We prove some theoretical results about existence and uniqueness of the solution of this abstract problem by using desirable functional spaces. We then apply a mixed and hybrid finite element method to discretize the transport equation by introducing appropriate basis functions. We present some numerical results to illustrate the efficiency of our method.
{"title":"Mixed Variational Formulation and Mixed-Hybrid Discretization of the Transport Equation","authors":"J. Cartier, M. Peybernes","doi":"10.1080/00411450.2010.529630","DOIUrl":"https://doi.org/10.1080/00411450.2010.529630","url":null,"abstract":"We introduce a new variational formulation of the transport equation based on a mixed form of the transport equation. We prove some theoretical results about existence and uniqueness of the solution of this abstract problem by using desirable functional spaces. We then apply a mixed and hybrid finite element method to discretize the transport equation by introducing appropriate basis functions. We present some numerical results to illustrate the efficiency of our method.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"1 - 46"},"PeriodicalIF":0.0,"publicationDate":"2010-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2010.529630","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-11-30DOI: 10.1080/00411450903404820
Derek J. Daniel
The stochastic dynamics of a neuronal spiking model in neuroscience, when viewed as a large simulated network, are known to reduce to the classic problem of solving the Fokker-Planck equation, or the equivalent Kolmogorov differential equation in probability theory, for the numerical evaluation of the statistical properties of neurons as a random injection of ion currents. The problem here, however, is that the initial condition for the Fokker-Planck equation is a Dirac delta function so the actual implementation of Delta functions that at the same time can attain numerical stability can become problematic in computational neuroscience. Therefore, in this brief communication, a computational method for implementing such an initial condition is suggested, which itself has led to an exact solution for this problem.
{"title":"Fokker-Planck Solution for a Neuronal Spiking Model","authors":"Derek J. Daniel","doi":"10.1080/00411450903404820","DOIUrl":"https://doi.org/10.1080/00411450903404820","url":null,"abstract":"The stochastic dynamics of a neuronal spiking model in neuroscience, when viewed as a large simulated network, are known to reduce to the classic problem of solving the Fokker-Planck equation, or the equivalent Kolmogorov differential equation in probability theory, for the numerical evaluation of the statistical properties of neurons as a random injection of ion currents. The problem here, however, is that the initial condition for the Fokker-Planck equation is a Dirac delta function so the actual implementation of Delta functions that at the same time can attain numerical stability can become problematic in computational neuroscience. Therefore, in this brief communication, a computational method for implementing such an initial condition is suggested, which itself has led to an exact solution for this problem.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"38 1","pages":"383 - 391"},"PeriodicalIF":0.0,"publicationDate":"2009-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903404820","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58913422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-11-30DOI: 10.1080/00411450903404846
E. Grycko
Debye temperature Θ D of a substance plays a role in Solid State Theory; its experimental determination however requires a sophisticated device such that the value of Θ D is not available for many substances. A statistical possibility of approximation of Θ D is presented for substances whose boiling point and molecular mass are known. This approximation is based on the notion of zero entropy temperature, which serves as a regressor; statistical considerations reveal a strong correlation between the regressor and Debye temperature. This result can be interpreted as a bridge between Fluid and Solid State Theories.
{"title":"On a Statistical Interrelation Between Boiling Point and Debye Temperature","authors":"E. Grycko","doi":"10.1080/00411450903404846","DOIUrl":"https://doi.org/10.1080/00411450903404846","url":null,"abstract":"Debye temperature Θ D of a substance plays a role in Solid State Theory; its experimental determination however requires a sophisticated device such that the value of Θ D is not available for many substances. A statistical possibility of approximation of Θ D is presented for substances whose boiling point and molecular mass are known. This approximation is based on the notion of zero entropy temperature, which serves as a regressor; statistical considerations reveal a strong correlation between the regressor and Debye temperature. This result can be interpreted as a bridge between Fluid and Solid State Theories.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"38 1","pages":"392 - 397"},"PeriodicalIF":0.0,"publicationDate":"2009-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903404846","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58913431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-11-30DOI: 10.1080/00411450903404705
D. Shulaia
The spectral representation of the linear multigroup transport problem is applied to two examples. In the first example, we obtain the dispersion relations, normalization coefficients, and eigenfunctions for any order N of scattering by using the eigenfunctions for isotropic scattering as the basis. In the second we obtain the dispersion relations, normalization coefficients, and eigenfunctions for N+1 order scattering by using the eigenfunctions for Nth order scattering as the basis. New identities relating quantities referring to different orders of scattering are obtained as well as identities involving spectral integrals and moments of eigenfunctions. Independent calculations are carried out to verify relations obtained using the spectral representation. In 1981, Kanal and Davies obtained similar results for the case of the one-velocity transport theory.
{"title":"Some Applications of a Spectral Representation of the Linear Multigroup Transport Problem","authors":"D. Shulaia","doi":"10.1080/00411450903404705","DOIUrl":"https://doi.org/10.1080/00411450903404705","url":null,"abstract":"The spectral representation of the linear multigroup transport problem is applied to two examples. In the first example, we obtain the dispersion relations, normalization coefficients, and eigenfunctions for any order N of scattering by using the eigenfunctions for isotropic scattering as the basis. In the second we obtain the dispersion relations, normalization coefficients, and eigenfunctions for N+1 order scattering by using the eigenfunctions for Nth order scattering as the basis. New identities relating quantities referring to different orders of scattering are obtained as well as identities involving spectral integrals and moments of eigenfunctions. Independent calculations are carried out to verify relations obtained using the spectral representation. In 1981, Kanal and Davies obtained similar results for the case of the one-velocity transport theory.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"38 1","pages":"347 - 382"},"PeriodicalIF":0.0,"publicationDate":"2009-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903404705","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58913366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-11-24DOI: 10.1080/00411450903372126
A. Kaşkaş, C. Tezcan
Transport theory methods can be applied to optical oceanography to solve forward and inverse problems. The combination of delta function representing forward and backward scattering with isotropic scattering is used to obtain scalar and plane irradiances for Henyey-Greenstein phase function. Once the irradiances are obtained, the apparent optical properties can be found analytically and numerically.
{"title":"An Application of Transport Theory in Optical Oceanography: The Estimation of the Apparent Optical Properties Using Henyey-Greenstein Phase Function","authors":"A. Kaşkaş, C. Tezcan","doi":"10.1080/00411450903372126","DOIUrl":"https://doi.org/10.1080/00411450903372126","url":null,"abstract":"Transport theory methods can be applied to optical oceanography to solve forward and inverse problems. The combination of delta function representing forward and backward scattering with isotropic scattering is used to obtain scalar and plane irradiances for Henyey-Greenstein phase function. Once the irradiances are obtained, the apparent optical properties can be found analytically and numerically.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"38 1","pages":"317 - 329"},"PeriodicalIF":0.0,"publicationDate":"2009-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903372126","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58913351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-11-24DOI: 10.1080/00411450903404796
S. Charfi, A. Intissar, A. Jeribi
This article considers a time-dependent rectilinear transport equation that was first studied in B. Montagnini and V. Pierpaoli (Transport Theory and Statistical Physics 1(1) (1971) 59–75). The associated transport operator is the infinitesimal generator of a C 0-semigroup, its spectrum is discrete, and there are only finitely many eigenvalues in each vertical strip. We show that the C 0-semigroup can be expanded by its generalized eigenvectors, and we assert its differentiability.
{"title":"Expansion of Solution in Terms of Generalized Eigenvectors for a Rectilinear Transport Equation","authors":"S. Charfi, A. Intissar, A. Jeribi","doi":"10.1080/00411450903404796","DOIUrl":"https://doi.org/10.1080/00411450903404796","url":null,"abstract":"This article considers a time-dependent rectilinear transport equation that was first studied in B. Montagnini and V. Pierpaoli (Transport Theory and Statistical Physics 1(1) (1971) 59–75). The associated transport operator is the infinitesimal generator of a C 0-semigroup, its spectrum is discrete, and there are only finitely many eigenvalues in each vertical strip. We show that the C 0-semigroup can be expanded by its generalized eigenvectors, and we assert its differentiability.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"38 1","pages":"330 - 345"},"PeriodicalIF":0.0,"publicationDate":"2009-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903404796","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58913412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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