One hundred and thirty years after the work of Ludwig Boltzmann on the interpretation of the irreversibility of physical phenomena, and one century after Einstein's formulation of Special Relativity, we are still ...One hundred and thirty years after the work of Ludwig Boltzmann on the interpretation of the irreversibility of physical phenomena, and one century after Einstein's formulation of Special Relativity, we are still not sure what we mean when we talk of “time” or “arrow of time”. We shall try to show that one source of this difficulty is our tendency to confuse, at least verbally, time and becoming, i.e. the course of time and the arrow of time, two concepts that the formalisms of modern physics are careful to distinguish. The course of time is represented by a time line that leads us to define time as the producer of duration. It is customary to place on this time line a small arrow that, ironically, must not be confused with the “arrow of time”. This small arrow is only there to indicate that the course of time is oriented, has a well-defined direction, even if this direction is arbitrary. The arrow of time, on the other hand, indicates the possibility for physical systems to experience, over the course of time, changes or transforma-tions that prevent them from returning to their initial state forever. Contrary to what the ex-pression “arrow of time” suggests, it is there-fore not a property of time itself but a property of certain physical phenomena whose dynamic is irreversible. By its very definition, the arrow of time presupposes the existence of a well- established course of time within which – in addition – certain phenomena have their own temporal orientation. We think that it is worth-while to emphasize the difference between sev-eral issues traditionally subsumed under the label “the problem of the direction of time”. If the expressions “course of time”, “direction of time” and “arrow of time” were better defined, systematically distinguished from one another and always used in their strictest sense, the debate about time, irreversibility and becoming in physics would become clearer.展开更多
In this paper, a simple model for a closed multiverse as a finite probability space is analyzed. For each moment of time on a discrete time-scale, only a finite number of states are possible and hence each possible un...In this paper, a simple model for a closed multiverse as a finite probability space is analyzed. For each moment of time on a discrete time-scale, only a finite number of states are possible and hence each possible universe can be viewed as a path in a huge but finite graph. By considering very general statistical assumptions, essentially originating from Boltzmann, we make the set of all such paths (the multiverse) into a probability space, and argue that under certain assumptions, the probability for a monotonic behavior of the entropy is enormously much larger then for a behavior with low entropy at both ends. The methods used are just very simple combinatorial ones, but the conclusion suggests that we may live in a multiverse which from a global point of view is completely time-symmetric in the sense that universes with Time’s Arrow directed forwards and backwards are equally probable. However, for an observer confined to just one universe, time will still be asymmetric.展开更多
In this paper, an alternative approach to cosmology is discussed. Rather than starting from the field equations of general relativity, one can investigate the probability space of all possible universes and try to dec...In this paper, an alternative approach to cosmology is discussed. Rather than starting from the field equations of general relativity, one can investigate the probability space of all possible universes and try to decide what kind of universe is the most probable one. Here two quite different models for this probability space are presented: the combinatorial model and the random curvature model. In addition, it is briefly discussed how these models could be applied to explain two fundamental problems of cosmology: Time’s Arrow and the accelerating expansion.展开更多
This paper characterizes the optimal solution of subjective expected utility with S-shaped utility function, by using the prospect theory （PT）. We also prove the existence of Arrow-Debreu equilibrium.
文摘One hundred and thirty years after the work of Ludwig Boltzmann on the interpretation of the irreversibility of physical phenomena, and one century after Einstein's formulation of Special Relativity, we are still not sure what we mean when we talk of “time” or “arrow of time”. We shall try to show that one source of this difficulty is our tendency to confuse, at least verbally, time and becoming, i.e. the course of time and the arrow of time, two concepts that the formalisms of modern physics are careful to distinguish. The course of time is represented by a time line that leads us to define time as the producer of duration. It is customary to place on this time line a small arrow that, ironically, must not be confused with the “arrow of time”. This small arrow is only there to indicate that the course of time is oriented, has a well-defined direction, even if this direction is arbitrary. The arrow of time, on the other hand, indicates the possibility for physical systems to experience, over the course of time, changes or transforma-tions that prevent them from returning to their initial state forever. Contrary to what the ex-pression “arrow of time” suggests, it is there-fore not a property of time itself but a property of certain physical phenomena whose dynamic is irreversible. By its very definition, the arrow of time presupposes the existence of a well- established course of time within which – in addition – certain phenomena have their own temporal orientation. We think that it is worth-while to emphasize the difference between sev-eral issues traditionally subsumed under the label “the problem of the direction of time”. If the expressions “course of time”, “direction of time” and “arrow of time” were better defined, systematically distinguished from one another and always used in their strictest sense, the debate about time, irreversibility and becoming in physics would become clearer.
文摘In this paper, a simple model for a closed multiverse as a finite probability space is analyzed. For each moment of time on a discrete time-scale, only a finite number of states are possible and hence each possible universe can be viewed as a path in a huge but finite graph. By considering very general statistical assumptions, essentially originating from Boltzmann, we make the set of all such paths (the multiverse) into a probability space, and argue that under certain assumptions, the probability for a monotonic behavior of the entropy is enormously much larger then for a behavior with low entropy at both ends. The methods used are just very simple combinatorial ones, but the conclusion suggests that we may live in a multiverse which from a global point of view is completely time-symmetric in the sense that universes with Time’s Arrow directed forwards and backwards are equally probable. However, for an observer confined to just one universe, time will still be asymmetric.
文摘In this paper, an alternative approach to cosmology is discussed. Rather than starting from the field equations of general relativity, one can investigate the probability space of all possible universes and try to decide what kind of universe is the most probable one. Here two quite different models for this probability space are presented: the combinatorial model and the random curvature model. In addition, it is briefly discussed how these models could be applied to explain two fundamental problems of cosmology: Time’s Arrow and the accelerating expansion.
文摘This paper characterizes the optimal solution of subjective expected utility with S-shaped utility function, by using the prospect theory （PT）. We also prove the existence of Arrow-Debreu equilibrium.
