We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β≥0, t > 0, the Gibbs probability measur...We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β≥0, t > 0, the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as dPβ,t/dP0= Zβ,t（0）(-1eβ∫0tδ0（xs）ds),（0.1）where Zβ,t（0）≡E0[eβ∫<sup>t0δ0（xs）ds]. This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper（Cranston and Molchanov, 2007）, we showed that for dimensions d≥3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior for β above this critical value. The critical value was determined by means of the spectral properties of the operator ? + βδ0, where ? is the discrete Laplacian on Zd. This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer. In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is dimension-dependent.展开更多
The moist atmosphere with occurring precipitation is considered to be a multiphase fluid composed of dry air, water vapor and hydrometeors. These compositions move with different velocities： they take a macroscopic m...The moist atmosphere with occurring precipitation is considered to be a multiphase fluid composed of dry air, water vapor and hydrometeors. These compositions move with different velocities： they take a macroscopic motion with a reference velocity and a relative motion with a velocity deviated from the reference velocity. The reference velocity can be chosen as the velocities of dry air, a gas mixture and the total air mixture. The budget equations of continuity and momentum are formulated in the three reference-velocity frames. It is shown that the resulting equations are dependent on the chosen reference velocity. The diffusive flux due to compositions moving with velocities deviated from the reference velocity and the internal sources due to the phase transitions of water substances result in additional source terms in continuity and momentum equations. A continuity equation of the total mass is conserved and free of diffusive flux divergence if the reference velocity is referred to the velocity of the total air mixture. However, continuity equations in the dry-air and gas- mixture frames are not conserved due to the mass diffusive flux divergence. The diffusive flux introduces additional source terms in the momentum equation. In the dry-air frame, the diffusive flux of water substances and the phase transitions of water substances contribute to the change of the total momentum. The additional sources of total momentum in the frame of a gas mixture are associated with the diffusive flux of hydrometeors, the phase transitions of hydrometeors and the gas- mixture diffusive flux. In the frame of total air mixture, the contribution to the total momentum comes from the diffusive flux of all atmospheric compositions instead of the phase transitions. The continuity and momentum equations derived here are more complicated than the traditional model equations. With increasing computing power, it becomes possible to simulate atmospheric processes with these sophisticated equations. It is helpful to the improvement of 展开更多
在5×10~2Hz 到5×10~8Hz 频率范围内测量了 Tb 改性的铌酸锶钡(SBN)晶体的室温介电常数,并以1×10~4Hz 的频率测量了这些介电常数从室温至居里温度以上的变化。对于 c 轴和 a 轴介电常数,都观测到了介电弛豫,但 c 轴介电...在5×10~2Hz 到5×10~8Hz 频率范围内测量了 Tb 改性的铌酸锶钡(SBN)晶体的室温介电常数,并以1×10~4Hz 的频率测量了这些介电常数从室温至居里温度以上的变化。对于 c 轴和 a 轴介电常数,都观测到了介电弛豫,但 c 轴介电弛豫频率明显低于 a 轴介电弛豫频率。对于不同的组份来说,介电弛豫频率随居里温度降低而降低。这些晶体的铁电—顺电相变是弥散性的,弥散程度随Tb 含量的增加而增高。展开更多
基金supported by National Science Foundation of USA (Grant Nos. DMS1007176 and DMS-0706928)
文摘We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β≥0, t > 0, the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as dPβ,t/dP0= Zβ,t（0）(-1eβ∫0tδ0（xs）ds),（0.1）where Zβ,t（0）≡E0[eβ∫<sup>t0δ0（xs）ds]. This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper（Cranston and Molchanov, 2007）, we showed that for dimensions d≥3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior for β above this critical value. The critical value was determined by means of the spectral properties of the operator ? + βδ0, where ? is the discrete Laplacian on Zd. This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer. In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is dimension-dependent.
基金Project supported by the National Basic Research Program of China(Grant No.2009CB421505)the Key Program of the Chinese Academy of Sciences(Grant No.KZZD-EW-05)+2 种基金the National Natural Sciences Foundation of China(Grant Nos.41175060,40930950,and 41005005)the Project of CAMS,China(Grant No.2011LASW-B15)the Spectial Scientific Research Fund of Meteorological Public Welfare of Ministry of Sciences and Technology,China(Grant No.GYHY200906004)
文摘The moist atmosphere with occurring precipitation is considered to be a multiphase fluid composed of dry air, water vapor and hydrometeors. These compositions move with different velocities： they take a macroscopic motion with a reference velocity and a relative motion with a velocity deviated from the reference velocity. The reference velocity can be chosen as the velocities of dry air, a gas mixture and the total air mixture. The budget equations of continuity and momentum are formulated in the three reference-velocity frames. It is shown that the resulting equations are dependent on the chosen reference velocity. The diffusive flux due to compositions moving with velocities deviated from the reference velocity and the internal sources due to the phase transitions of water substances result in additional source terms in continuity and momentum equations. A continuity equation of the total mass is conserved and free of diffusive flux divergence if the reference velocity is referred to the velocity of the total air mixture. However, continuity equations in the dry-air and gas- mixture frames are not conserved due to the mass diffusive flux divergence. The diffusive flux introduces additional source terms in the momentum equation. In the dry-air frame, the diffusive flux of water substances and the phase transitions of water substances contribute to the change of the total momentum. The additional sources of total momentum in the frame of a gas mixture are associated with the diffusive flux of hydrometeors, the phase transitions of hydrometeors and the gas- mixture diffusive flux. In the frame of total air mixture, the contribution to the total momentum comes from the diffusive flux of all atmospheric compositions instead of the phase transitions. The continuity and momentum equations derived here are more complicated than the traditional model equations. With increasing computing power, it becomes possible to simulate atmospheric processes with these sophisticated equations. It is helpful to the improvement of
文摘在5×10~2Hz 到5×10~8Hz 频率范围内测量了 Tb 改性的铌酸锶钡(SBN)晶体的室温介电常数,并以1×10~4Hz 的频率测量了这些介电常数从室温至居里温度以上的变化。对于 c 轴和 a 轴介电常数,都观测到了介电弛豫,但 c 轴介电弛豫频率明显低于 a 轴介电弛豫频率。对于不同的组份来说,介电弛豫频率随居里温度降低而降低。这些晶体的铁电—顺电相变是弥散性的,弥散程度随Tb 含量的增加而增高。
