The usual phenomenological laws of matter, like equations of state and
transport equations, are a... more The usual phenomenological laws of matter, like equations of state and transport equations, are average laws. They deal in macroscopic variables like pressure, temperature, heat flux and electrical current, which represent the aggregate effect of myriad molecular interactions. Likewise, the physical parameters appearing in the laws (compressibility, friction constant) characterize the behaviour of materials on a macroscopic scale, and represent molecular quantities only implicitly, as specialized indices of their overall, collective effect. The equations themselves derive from the underlying laws of molecular motion by averaging procedures applied to the Liouville equation, which governs the dynamical evolution of the complete set of molecular coordinates (though few such derivations have actually been carried out, and none of the laws were found in this way). The idea of fluctuation theory is to refine the description such phenomenological laws provide by treating them literally as average equations, and seeking probabilistic laws for the macroscopic variables. From this point of view the usual macroscopic law only describes how the mean values behave, and one seeks the probability distribution of actual values around the mean.
For systems not close to equilibrium, governed by nonlinear laws, there is yet no general phenomenological theory of fluctuations like the Onsager - Machlup generalization of Langevin’s theory of Brownian motion. A few such problems have been solved on a microscopic basis starting from a master equation, including the extension of the nonlinear Boltzmann equation for fluctuations arbitrarily far from equilibrium for a homogeneous dilute gas. Here, we restrict ourselves entirely to fluctuation problems involving simple dilute gases.
Based in part on a series of lectures given at L'Ecole Polytechnique Federate at Lausanne, Switzerland as a part of their Troisieme Cycle, Summer 1974.
The evolution of a homogeneous dilute gas is treated as a Markov process in the complete set of K... more The evolution of a homogeneous dilute gas is treated as a Markov process in the complete set of K coarse-grained velocity states of all N particles. From the Siegert master equation for the process, a Fokker-Planck equation is derived which describes, in the limit N → ∞, the fluctuations in the occupation numbers ni(t), whose average behavior is governed by the (appropriately discretized) Boltzmann equation. The continuum limit K → ∞ corresponds to fluctuations in the usual molecular distribution function f(r v;t). On similar reasoning, a Fokker-Planck equation is obtained for the fluctuation process near equilibrium, where the average is governed by the linearized Boltzmann equation. The theory of linear irreversible processes, which offers a statistical description of fluctuations on a thermodynamical basis, is applied to the linearized Boltzmann equation—treated as a linear phenomenological equation—following the development given recently by Fox and Uhlenbeck. The resulting stochastic equation is seen to be equivalent to the Fokker- Planck equation obtained from the master equation, yielding a multidimensional Ornstein-Uhlenbeck process which describes the fluctuations in molecular phase space.
As Allied scientists established the feasibility of an atomic bomb, Germany's leading theorist ca... more As Allied scientists established the feasibility of an atomic bomb, Germany's leading theorist came to a mistaken conclusion Jonothan Logan Not far into the 200-odd pages of the recently declassified Farm Hall transcripts comes the extraordinary mo? ment when Werner Heisenberg and the nine other German nuclear scientists being held at a country house in Eng? land hear that an Allied Atomic bomb has devastated the city of Hiroshima. The Allies' atomic weapon, according to the BBC Home Service announcer, has delivered "as much explosive power as 2,000 of our great ten-tonners." Hidden microphones conveyed the voices of the German "guests" to a nearby Hstening room at Farm Hall, and so we have in the reports a record of their astonished reaction to this news. Heisenberg re? sponds by flatly rejecting the possibility that the bomb could have been a fission weapon. "Some dilettante in America who knows very little about it has bluffed them," he says. "I don't believe that it has anything to do with uranium." Even more informative, from a scien? tist's point of view, are the intense tech? nical discussions that consumed the subsequent hours and days as the cap? tive scientists tried to puzzle out how their Allied counterparts could have managed to do what they had conclud? ed was beyond reach. Of particular in? terest, and the main subject of this arti? cle, is Heisenberg's informal estimate of the amount of uranium required for a bomb?the critical mass. Throughout the war Heisenberg seems to have be? lieved that many tons of the rare isotope Jonathan Logan was trained as a theoretical physi? cist a t the University of Chicago and Rockefeller University, then turned to molecular genetics as a postdoctoral fellow at Harvard. Before that he was a member of the editorial staff of the Physical Review, where an association with the journal's
Examines in detail the scientific background of the Allied and German wartime fission programs, p... more Examines in detail the scientific background of the Allied and German wartime fission programs, particularly as they shaped differing conceptions of the feasibility of developing an atomic bomb. Parallel scientific time tables are provided along with a close analysis of the mistaken critical mass calculation outlined by Werner Heisenberg upon hearing of the successful Allied bomb in 1945. 35 refs., 11 figs., 5 tabs
No one better represents the plight and the conduct of German intellectuals under Hitler than Wer... more No one better represents the plight and the conduct of German intellectuals under Hitler than Werner Heisenberg, whose task it was to build an atomic bomb for Nazi Germany. The controversy surrounding Heisenberg still rages, because of the nature of his work and the regime for which it was undertaken. What precisely did Heisenberg know about the physics of the atomic bomb? How deep was his loyalty to the German government during the Third Reich? Assuming that he had been able to build a bomb, would he have been willing? These questions, the moral and the scientific, are answered by Paul Lawrence Rose with greater accuracy and breadth of documentation than any other historian has yet achieved. Digging deep into the archival record among formerly secret technical reports, Rose establishes that Heisenberg never overcame certain misconceptions about nuclear fission, and as a result the German leaders never pushed for atomic weapons. In fact, Heisenberg never had to face the moral problem of whether he should design a bomb for the Nazi regime. Only when he and his colleagues were interned in England and heard about Hiroshima did Heisenberg realize that his calculations were wrong. He began at once to construct an image of himself as a 'pure' scientist who could have built a bomb but chose to work on reactor design instead. This was fiction, as Rose demonstrates: in reality, Heisenberg blindly supported and justified the cause of German victory. The question of why he did, and why he misrepresented himself afterwards, is answered through Rose's subtle analysis of German mentality and the scientists' problems of delusion and self-delusion. This fascinating study is a profound effort to understand one of the twentieth century's great enigmas.
The usual phenomenological laws of matter, like equations of state and
transport equations, are a... more The usual phenomenological laws of matter, like equations of state and transport equations, are average laws. They deal in macroscopic variables like pressure, temperature, heat flux and electrical current, which represent the aggregate effect of myriad molecular interactions. Likewise, the physical parameters appearing in the laws (compressibility, friction constant) characterize the behaviour of materials on a macroscopic scale, and represent molecular quantities only implicitly, as specialized indices of their overall, collective effect. The equations themselves derive from the underlying laws of molecular motion by averaging procedures applied to the Liouville equation, which governs the dynamical evolution of the complete set of molecular coordinates (though few such derivations have actually been carried out, and none of the laws were found in this way). The idea of fluctuation theory is to refine the description such phenomenological laws provide by treating them literally as average equations, and seeking probabilistic laws for the macroscopic variables. From this point of view the usual macroscopic law only describes how the mean values behave, and one seeks the probability distribution of actual values around the mean.
For systems not close to equilibrium, governed by nonlinear laws, there is yet no general phenomenological theory of fluctuations like the Onsager - Machlup generalization of Langevin’s theory of Brownian motion. A few such problems have been solved on a microscopic basis starting from a master equation, including the extension of the nonlinear Boltzmann equation for fluctuations arbitrarily far from equilibrium for a homogeneous dilute gas. Here, we restrict ourselves entirely to fluctuation problems involving simple dilute gases.
Based in part on a series of lectures given at L'Ecole Polytechnique Federate at Lausanne, Switzerland as a part of their Troisieme Cycle, Summer 1974.
The evolution of a homogeneous dilute gas is treated as a Markov process in the complete set of K... more The evolution of a homogeneous dilute gas is treated as a Markov process in the complete set of K coarse-grained velocity states of all N particles. From the Siegert master equation for the process, a Fokker-Planck equation is derived which describes, in the limit N → ∞, the fluctuations in the occupation numbers ni(t), whose average behavior is governed by the (appropriately discretized) Boltzmann equation. The continuum limit K → ∞ corresponds to fluctuations in the usual molecular distribution function f(r v;t). On similar reasoning, a Fokker-Planck equation is obtained for the fluctuation process near equilibrium, where the average is governed by the linearized Boltzmann equation. The theory of linear irreversible processes, which offers a statistical description of fluctuations on a thermodynamical basis, is applied to the linearized Boltzmann equation—treated as a linear phenomenological equation—following the development given recently by Fox and Uhlenbeck. The resulting stochastic equation is seen to be equivalent to the Fokker- Planck equation obtained from the master equation, yielding a multidimensional Ornstein-Uhlenbeck process which describes the fluctuations in molecular phase space.
As Allied scientists established the feasibility of an atomic bomb, Germany's leading theorist ca... more As Allied scientists established the feasibility of an atomic bomb, Germany's leading theorist came to a mistaken conclusion Jonothan Logan Not far into the 200-odd pages of the recently declassified Farm Hall transcripts comes the extraordinary mo? ment when Werner Heisenberg and the nine other German nuclear scientists being held at a country house in Eng? land hear that an Allied Atomic bomb has devastated the city of Hiroshima. The Allies' atomic weapon, according to the BBC Home Service announcer, has delivered "as much explosive power as 2,000 of our great ten-tonners." Hidden microphones conveyed the voices of the German "guests" to a nearby Hstening room at Farm Hall, and so we have in the reports a record of their astonished reaction to this news. Heisenberg re? sponds by flatly rejecting the possibility that the bomb could have been a fission weapon. "Some dilettante in America who knows very little about it has bluffed them," he says. "I don't believe that it has anything to do with uranium." Even more informative, from a scien? tist's point of view, are the intense tech? nical discussions that consumed the subsequent hours and days as the cap? tive scientists tried to puzzle out how their Allied counterparts could have managed to do what they had conclud? ed was beyond reach. Of particular in? terest, and the main subject of this arti? cle, is Heisenberg's informal estimate of the amount of uranium required for a bomb?the critical mass. Throughout the war Heisenberg seems to have be? lieved that many tons of the rare isotope Jonathan Logan was trained as a theoretical physi? cist a t the University of Chicago and Rockefeller University, then turned to molecular genetics as a postdoctoral fellow at Harvard. Before that he was a member of the editorial staff of the Physical Review, where an association with the journal's
Examines in detail the scientific background of the Allied and German wartime fission programs, p... more Examines in detail the scientific background of the Allied and German wartime fission programs, particularly as they shaped differing conceptions of the feasibility of developing an atomic bomb. Parallel scientific time tables are provided along with a close analysis of the mistaken critical mass calculation outlined by Werner Heisenberg upon hearing of the successful Allied bomb in 1945. 35 refs., 11 figs., 5 tabs
No one better represents the plight and the conduct of German intellectuals under Hitler than Wer... more No one better represents the plight and the conduct of German intellectuals under Hitler than Werner Heisenberg, whose task it was to build an atomic bomb for Nazi Germany. The controversy surrounding Heisenberg still rages, because of the nature of his work and the regime for which it was undertaken. What precisely did Heisenberg know about the physics of the atomic bomb? How deep was his loyalty to the German government during the Third Reich? Assuming that he had been able to build a bomb, would he have been willing? These questions, the moral and the scientific, are answered by Paul Lawrence Rose with greater accuracy and breadth of documentation than any other historian has yet achieved. Digging deep into the archival record among formerly secret technical reports, Rose establishes that Heisenberg never overcame certain misconceptions about nuclear fission, and as a result the German leaders never pushed for atomic weapons. In fact, Heisenberg never had to face the moral problem of whether he should design a bomb for the Nazi regime. Only when he and his colleagues were interned in England and heard about Hiroshima did Heisenberg realize that his calculations were wrong. He began at once to construct an image of himself as a 'pure' scientist who could have built a bomb but chose to work on reactor design instead. This was fiction, as Rose demonstrates: in reality, Heisenberg blindly supported and justified the cause of German victory. The question of why he did, and why he misrepresented himself afterwards, is answered through Rose's subtle analysis of German mentality and the scientists' problems of delusion and self-delusion. This fascinating study is a profound effort to understand one of the twentieth century's great enigmas.
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Papers by Jonothan Logan
transport equations, are average laws. They deal in macroscopic variables
like pressure, temperature, heat flux and electrical current, which represent
the aggregate effect of myriad molecular interactions. Likewise, the physical parameters appearing in the laws (compressibility, friction constant) characterize the behaviour of materials on a macroscopic scale, and represent molecular quantities only implicitly, as specialized indices of their overall, collective effect.
The equations themselves derive from the underlying laws of molecular
motion by averaging procedures applied to the Liouville equation, which
governs the dynamical evolution of the complete set of molecular
coordinates (though few such derivations have actually been carried out,
and none of the laws were found in this way). The idea of fluctuation theory is to refine the description such phenomenological laws provide by treating them literally as average equations, and seeking probabilistic laws for the macroscopic variables. From this point of view the usual macroscopic law only describes how the mean values behave, and one seeks the probability distribution of actual values around the mean.
For systems not close to equilibrium, governed by nonlinear laws, there is
yet no general phenomenological theory of fluctuations like the Onsager -
Machlup generalization of Langevin’s theory of Brownian motion. A few
such problems have been solved on a microscopic basis starting from a
master equation, including the extension of the nonlinear Boltzmann
equation for fluctuations arbitrarily far from equilibrium for a homogeneous dilute gas. Here, we restrict ourselves entirely to fluctuation problems involving simple dilute gases.
Based in part on a series of lectures given at L'Ecole Polytechnique Federate at Lausanne, Switzerland as a part of their Troisieme Cycle, Summer 1974.
Boltzmann equation—treated as a linear phenomenological equation—following the development given recently by Fox and Uhlenbeck. The resulting stochastic equation is seen to be equivalent to the Fokker-
Planck equation obtained from the master equation, yielding a multidimensional Ornstein-Uhlenbeck process which describes the fluctuations in molecular phase space.
transport equations, are average laws. They deal in macroscopic variables
like pressure, temperature, heat flux and electrical current, which represent
the aggregate effect of myriad molecular interactions. Likewise, the physical parameters appearing in the laws (compressibility, friction constant) characterize the behaviour of materials on a macroscopic scale, and represent molecular quantities only implicitly, as specialized indices of their overall, collective effect.
The equations themselves derive from the underlying laws of molecular
motion by averaging procedures applied to the Liouville equation, which
governs the dynamical evolution of the complete set of molecular
coordinates (though few such derivations have actually been carried out,
and none of the laws were found in this way). The idea of fluctuation theory is to refine the description such phenomenological laws provide by treating them literally as average equations, and seeking probabilistic laws for the macroscopic variables. From this point of view the usual macroscopic law only describes how the mean values behave, and one seeks the probability distribution of actual values around the mean.
For systems not close to equilibrium, governed by nonlinear laws, there is
yet no general phenomenological theory of fluctuations like the Onsager -
Machlup generalization of Langevin’s theory of Brownian motion. A few
such problems have been solved on a microscopic basis starting from a
master equation, including the extension of the nonlinear Boltzmann
equation for fluctuations arbitrarily far from equilibrium for a homogeneous dilute gas. Here, we restrict ourselves entirely to fluctuation problems involving simple dilute gases.
Based in part on a series of lectures given at L'Ecole Polytechnique Federate at Lausanne, Switzerland as a part of their Troisieme Cycle, Summer 1974.
Boltzmann equation—treated as a linear phenomenological equation—following the development given recently by Fox and Uhlenbeck. The resulting stochastic equation is seen to be equivalent to the Fokker-
Planck equation obtained from the master equation, yielding a multidimensional Ornstein-Uhlenbeck process which describes the fluctuations in molecular phase space.