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不同均相流模型均相流模型是用于描述多相流系统中的均匀性或各向同性的假设。
这种模型在物理上可以应用于流体力学、化学反应动力学和热力学等领域。
根据不同的应用场景和需求，有多种不同的均相流模型，以下是其中几种常见的模型：1.理想均相流模型理想均相流模型假设多相流系统中的各相是均匀分布的，没有相分离或聚集的现象。
各相之间的界面消失，整个系统具有均匀的物理性质。
这种模型通常适用于描述流体动力学中的均匀流动，如不可压缩的牛顿流体。
2.分子均相流模型分子均相流模型基于分子动力学理论，将多相流系统视为由分子组成，并考虑分子之间的相互作用和碰撞。
这种模型可以准确地描述微观尺度上的流动现象，如粘性、扩散和传热等。
然而，由于计算复杂度高，分子均相流模型在实际工程中的应用受到限制。
3.统计均相流模型统计均相流模型结合了分子动力学和宏观流体动力学的方法，通过引入概率密度函数来描述多相流系统中分子的分布和运动状态。
这种模型可以处理分子之间的相互作用和统计涨落效应，同时保持了宏观流体动力学的简单性和准确性。
4.连续介质均相流模型连续介质均相流模型假设多相流系统中的各相是连续分布的，没有明显的界面或颗粒。
这种模型通常适用于描述宏观尺度上的流动现象，如管道流动、涡旋等。
连续介质均相流模型可以进一步细分为无粘性流动、粘性流动和扩散流动等不同类型。
5.多尺度均相流模型多尺度均相流模型考虑了多相流系统中不同尺度的流动现象，包括微观尺度、宏观尺度和介观尺度等。
这种模型通过建立不同尺度之间的联系和转换关系，综合利用各个尺度的动力学机制，以获得更全面和准确的流动特性。
多尺度均相流模型在多相流系统的模拟和分析中具有广泛的应用前景。
这些不同的均相流模型在数学表达式、物理意义和应用范围方面存在差异。
在实际应用中，需要根据具体问题的需求选择合适的均相流模型，并进行相应的参数拟合或调整。
同时，需要注意模型的局限性，对于超出适用范围的问题，可能需要采用更复杂的数值模拟方法或实验研究手段来处理。
数学mathematics公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis, hypotheses(pl.)命题proposition算术arithmetic加plus(prep.), add(v.), addition(n.) 被加数augend, summand加数addend和sum减minus(prep.), subtract(v.), subtraction(n.)被减数minuend减数subtrahend差remainder乘times(prep.), multiply(v.), multiplication(n.)被乘数multiplicand, faciend乘数multiplicator积product除divided by(prep.), divide(v.), division(n.)被除数dividend除数divisor商quotient等于equals, is equal to, is equivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator数字digit数number自然数natural number整数integer小数decimal小数点decimal point 分数fraction分子numerator分母denominator比ratio正positive负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae(pl.)单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor, z-factor等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth powern次方the power of n, the nth power 开方evolution, extraction二次方根，平方根square root三次方根，立方根cube root四次方根the root of four, the fourth rootn次方根the root of n, the nth root 集合aggregate元素element空集void子集subset交集intersection并集union补集complement映射mapping函数function定义域domain, field of definition值域range常量constant变量variable单调性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无穷大infinite(a.) infinity(n.)无穷小infinitesimal积分integral定积分definite integral不定积分indefinite integral有理数rational number无理数irrational number实数real number虚数imaginary number复数complex number矩阵matrix 行列式determinant几何geometry点point线line 面plane体solid线段segment射线radial平行parallel相交intersect角angle角度degree弧度radian锐角acute angle直角right angle钝角obtuse angle平角straight angle周角perigon底base边side高height三角形triangle锐角三角形acute triangle直角三角形right triangle直角边leg斜边hypotenuse勾股定理Pythagorean theorem钝角三角形obtuse triangle不等边三角形scalene triangle等腰三角形isosceles triangle等边三角形equilateral triangle四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width菱形rhomb, rhombus, rhombi(pl.), diamond正方形square梯形trapezoid直角梯形right trapezoid等腰梯形isosceles trapezoid五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon十一边形hendecagon十二边形dodecagon多边形polygon正多边形equilateral polygon 圆circle圆心center半径radius直径diameter圆周率pi弧arc半圆semicircle扇形sector环ring椭圆ellipse圆周circumference周长perimeter面积area轨迹locus, loca(pl.)相似similar全等congruent四面体tetrahedron五面体pentahedron六面体hexahedron平行六面体parallelepiped 立方体cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体hendecahedron十二面体dodecahedron二十面体icosahedron多面体polyhedron棱锥pyramid棱柱prism棱台frustum of a prism旋转rotation轴axis 圆锥cone圆柱cylinder圆台frustum of a cone球sphere半球hemisphere底面undersurface表面积surface area体积volume空间space坐标系coordinates坐标轴x-axis, y-axis, z-axis 横坐标x-coordinate纵坐标y-coordinate原点origin双曲线hyperbola抛物线parabola三角trigonometry正弦sine余弦cosine正切tangent余切cotangent正割secant余割cosecant反正弦arc sine反余弦arc cosine反正切arc tangent反余切arc cotangent反正割arc secant反余割arc cosecant相位phase周期period振幅amplitude内心incenter外心excenter旁心escenter垂心Orthocenter重心barycenter内切圆inscribed circle外切圆circumcircle统计statistics平均数average加权平均数weighted average方差variance标准差root-mean-square deviation, standard deviation比例proportion百分比percent百分点percentage百分位数percentile排列permutation组合combination概率，或然率probability分布distribution正态分布normal distribution非正态分布abnormal distribution 图表graph条形统计图bar graph柱形统计图histogram折线统计图broken line graph曲线统计图curve diagram扇形统计图pie diagram。
——名词解释将因变量与一组解释变量和未观测到的扰动联系起来的方程，方程中未知的总体参数决定了各解释变量在其他条件不变下的效应。
与经济分析不同，在进行计量经济分析之前，要明确变量之间的函数形式。
经验分析（Empirical Analysis）：在规范的计量分析中，用数据检验理论、估计关系式或评价政策有效性的研究。
确定遗漏变量、测量误差、联立性或其他某种模型误设所导致的可能偏误的过程线性概率模型（LPM）（Linear Probability Model, LPM）：响应概率对参数为线性的二值响应模型。
没有一个模型可以通过对参数施加限制条件而被表示成另一个模型的特例的两个（或更多）模型。
有限分布滞后（FDL）模型（Finite Distributed Lag (FDL) Model）：允许一个或多个解释变量对因变量有滞后效应的动态模型。
布罗施-戈弗雷检验（Breusch-Godfrey Test）：渐近正确的AR（p）序列相关检验，以AR（1）最为流行；该检验考虑到滞后因变量和其他不是严格外生的回归元。
布罗施-帕甘检验（Breusch-Pagan Test）/（BP Test）：将OLS 残差的平方对模型中的解释变量做回归的异方差性检验。
若一个模型正确，则另一个非嵌套模型得到的拟合值在该模型是不显著的。
因此，这是相对于非嵌套对立假设而对一个模型的检验。
在模型中包含对立模型的拟合值，并使用对拟合值的t 检验来实现。
回归误差设定检验（RESET）（Regression Specification Error Test, RESET）：在多元回归模型中，检验函数形式的一般性方法。
它是对原OLS 估计拟合值的平方、三次方以及可能更高次幂的联合显著性的F 检验。
怀特检验（White Test）：异方差的一种检验方法，涉及到做OLS 残差的平方对OLS 拟合值和拟合值的平方的回归。
这种检验方法的最一般的形式是，将OLS 残差的平方对解释变量、解释变量的平方和解释变量之间所有非多余的交互项进行回归。
Phase-field 方法的基本思路\特点摘要：本篇文章的主要内容是关于phase-field方法的，介绍了相场法的基本思想，简述了相场法的计算方法及优缺点，还对相场法的应用情况作了概述。
关键字：相场法两相计算方法应用Abstract: The main content of this article is about the phase-field method, which introduces the basic idea of phase field method, the calculation method and the advantages and disadvantages of the phase field method, and an overview of the application of the phase field method.Key words: phase field method; two-phase flow; calculation method; application 1相场法的基本思想相场方法是以金兹堡—朗道理论为基础，用微分方程来体现扩散、有序化势和热力学驱动的综合作用。
相场是一种计算技术，它能使研究者直接模拟微观组织的形成。
因此，相场法也称为直接的微观组织模拟。
与其他方法不同，相场法引入了相场变量这个概念来描述系统在空间/时间上每个位置的物理状态。
相场变量用φ(r,t)表示，其中r代表空间，t代表时间，在液相区，φ(r,t)=-1；在固相区φ(r,t)=1；而在固液两相区，φ(r,t)的值在-1~1之间变化。
2相场模拟中的计算方法2.1有限差分法有限差分法的发展已比较的成熟，并且十分重要的是它的程序结构十分简单。
人们相继实现了对枝晶生长定性和定量模拟，并且与预测结果十分吻合。
在国内，分别实现了对镍的等轴枝晶生长、铝的枝晶生长和对铝铜合金的生长过程的模拟。
简述菲德勒模型一、引言菲德勒模型是一种用于描述动态系统的数学模型，由美国物理学家Edward Lorenz于1963年提出。
该模型被广泛应用于气象学、生态学、经济学等领域，并成为了混沌理论的基础之一。
本文将对菲德勒模型进行全面详细的简述。
二、菲德勒模型的基本概念1. 状态空间：菲德勒模型中，状态空间是指系统可能处于的所有状态所组成的空间。
2. 相空间：相空间是指在状态空间中，由系统所有状态所组成的集合。
3. 相轨道：相轨道是指系统从一个初始状态开始，在相空间中运动形成的轨迹。
三、菲德勒模型的数学表达式1. 菲德勒方程：菲德勒方程是描述大气运动规律的微分方程组，它包含三个变量：x、y和z。
其表达式为：dx/dt = σ(y-x)dy/dt = x(ρ-z)-ydz/dt = xy-βz其中，σ、ρ和β分别代表大气运动中涉及到的参数。
2. 菲德勒模型：将菲德勒方程简化为如下形式：x(t+1) = y(t) + a[x(t)-y(t)]y(t+1) = x(t) + b[x(t)-y(t)]其中，a和b是常数。
四、菲德勒模型的特点1. 敏感依赖初值：菲德勒模型中，微小的初始条件变化会导致系统的行为发生巨大的不同。
2. 非周期性：菲德勒模型中，相轨道不会重复出现相同的路径。
3. 空间混沌性：菲德勒模型中，相空间中的轨道具有混沌性质，即无法准确预测系统未来状态。
五、菲德勒模型在气象学中的应用1. 气候预测：通过对大气运动规律进行建模，可以预测未来一段时间内的气候变化趋势。
2. 暴雨预警：利用菲德勒模型对大气运动进行建模，可以提前发现暴雨等极端天气事件，并及时采取措施减少灾害损失。
3. 空气污染监测：通过对大气运动规律进行建模，可以监测和预测空气污染物在不同地区之间的传播和扩散情况。
六、菲德勒模型在其他领域的应用1. 生态学：菲德勒模型可以用于研究生态系统中物种数量和种群变化趋势。
2. 经济学：菲德勒模型可以用于研究经济系统中的市场波动和价格变化趋势。
费米狄拉克分布平均场近似费米狄拉克分布是描述费米子统计的一种概率分布。
费米子是一类具有半整数自旋的粒子，如电子、质子和中子等。
费米狄拉克分布描述的是以费米-狄拉克统计为基础的系统中，不可区分自旋1/2费米粒子的分布概率。
平均场近似是一种常用的近似方法，通过将复杂的相互作用问题简化为平均场问题来求解。
在费米-狄拉克系统中，通过平均场近似可以得到粒子的平均分布情况，从而揭示系统的宏观行为。
费米狄拉克分布的表达式如下：f(E) = 1 / (exp((E - Ef) / kT) + 1)其中，f(E)是能级E上的粒子占据数，Ef是费米能级，k是玻尔兹曼常数，T是系统的温度。
费米狄拉克分布的特点有两个：一是它是一个分段函数，对于能级E小于费米能级Ef的情况，粒子的占据数为1，表示该能级已被全部占据；对于能级E大于费米能级Ef的情况，粒子的占据数为0，表示该能级无粒子占据。
二是随着温度的升高，费米分布在能级E附近的概率密度会变得更加平坦，即费米子更容易占据较高能级。
在具体的物理问题中，我们经常需要计算物理量的平均值。
在费米狄拉克分布中，我们可以通过对能级加权求和的方式计算平均值。
对于某一物理量A，其平均值可以表示为：<A> = ΣA*f(E)其中，Σ表示对所有能级求和。
对于复杂的系统，一般很难得到能级的解析表达式，因此我们需要使用平均场近似来求解。
平均场近似假设系统中的每个粒子都只受到平均场的影响，而忽略其他粒子对其的相互作用。
在费米-狄拉克系统中，平均场近似意味着每个粒子都只受到平均电势的影响，而忽略其他粒子对其的库仑相互作用。
通过平均场近似，我们可以将费米-狄拉克分布简化为下面的形式：f(E) = 1 / (exp((E - U) / kT) + 1)其中，U是平均电势。
平均电势可以通过求解平均场方程来得到，平均场方程是一个自洽方程：U = ΣE*f(E)平均场方程描述了粒子的分布和平均电势之间的关系。
a rXiv:g r-qc/98811v14A ug1998Phase diagram of the mean field model of simplicial gravity P.Bialas a ∗,Z.Burda b and D.Johnston c a Fakult¨a t fur Physik,Universit¨a t Bielefeld,33501Bielefeld Germany b Institute of Physics,Jagellonian University,30-059Krakow,Poland c Dept.of Mathematics,Heriot–Watt University,EH144AS Edinburgh,Scotland Abstract We discuss the phase diagram of the balls in boxes model,with a varying number of boxes.The model can be regarded as a mean-field model of simplicial gravity.We analyse in detail the case of weights of the form p (q )=q −β,which correspond to the measure term introduced in the simplicial quantum gravity simulations.The system has two phases :elongated (fluid )and crumpled .For β∈(2,∞)the transition between these two phases is first order,while for β∈(1,2]it is continuous.The transition becomes softer when βapproaches unity and eventually disappears at β=1.We then generalise the discussionto an arbitrary set of weights.Finally,we show that if one introduces an additional kinematic bound on the average density of balls per box then a new condensed phase appears in the phase diagram.It bears some similarity to the crinkled phase of simplicial gravity discussed recently in models of gravity interacting with matter fields.The mean field description of simplicial gravity introduced in [1,2,3]was analysed in the series of papers [4,5,6].As discussed there,the mean field approximation explains many facts observed in numerical experiments of simplicial gravity,such as the appearance of singular vertices and themother universe,and the discontinuity of the phase transition.Recent work [7,8]shows that adding vector matterfields to simplicial gravity effectively amounts to renormalising theαparameter in the measure term qαwhere q is the vertex order.The resulting phase diagram resembles the phase diagram of the mean–field model which we discuss in detail in this letter.The model is given by the partition function[3]:Z(N,κ,β)=M maxM=1eκM q1,...,q M p(q1)···p(q M)δq1+···+q M,N(1)=M maxM=1eκM z(N,M,β)(2)It describes an ensemble of N balls distributed in a varying number of boxes, M.The system has at least one box and at most M max boxes.The partitions of balls are weighted by the product of one-box weights p(q),where q is the number of balls in the box.Here we consider one parameter family of weights:p(q)=q−βfor q=1,2, (3)Note that for these weights the minimal number of balls in a box is q min=1. If there are no further constraints this implies that the maximal number of boxes is equal to the number of balls M max=N.For large N the partition functions z(...)and Z(...)are expected to behave as:z(N,M,β)∼N y e Nf(r,β)with r=MThe additional factor N on the right hand side of(6)comes from the inte-gration measure Ndr.In the thermodynamic limit,this gives:φ(κ,β)=f(r∗,β)+r∗κ.(7) where r∗is maximum of the integrand in(6).The value of r∗is given by the equation:−κ=∂r f(r∗,β)(8) if the maximum of the integrand lies inside the interval(0,1)and is equal r∗=0or r∗=1otherwise.Additionally we see that in the case when the maximum lies inside the interval(0,1)the integration over r around the saddle point introduces the additional factor N−1/2.This leads to the following relation between the exponents y and Y:Y=y+1N =1Z(N,κ,β).(11)As we will see this quantity plays the role of the order parameter.In the thermodynamic limit the equation(11)reads:r =∂κφ(κ,β)=r∗.(12) The second equality in the last formula can be treated as an inverse transform to(8).It results from the fact that the integrand of(6),which corresponds to the distribution of r,is sharply peaked around r∗when N→∞and the position of the peak also becomes the average of the distribution.The task of determining r∗relies onfinding the maximum of the function f(r,β)+κr.The function f(r,β)can be found by the saddle point method [2]:f(r,β)=µ∗(r)+rK(µ∗(r),β)(13)3whereµ∗(r)is a value which maximises the right hand side of the above expression for the given r.The function K(µ,β)is defined as:K(µ,β)=log∞q=1p(q)e−µq.(14)The properties of the function K(µ,β)are central to the further consider-ations as they fully determine the behaviour of the model.For the particular choice of weights(3):K(µ,β)=log∞q=1q−βe−µq=log gβ(µ)(15)where gβ(µ)is a function familiar from Bose-Einstein condensation,which has an integral representation:gβ(µ)=1eµ+t−1.(16)The function gβ(µ)is defined on the interval[0,∞),soµ∗is either a solution of the saddle point equation:1gβ(µ∗)(17) for r>r cr or elseµ∗=0.The critical value of r isr cr= 0β≤2ζ(β)0.00.20.40.60.8 1.0r 0.000.100.200.300.400.00.20.40.60.8 1.0−4.0−3.0−2.0−1.00.01.02.0−4.0−3.0−2.0−1.00.01.02.03.04.0κ0.00.10.20.30.40.50.60.70.80.91.0r *Figure 3:The value of r ∗(κ)for various values of β.The portions of plots lying on the x axis (r ∗=0)are not plotted.line is a discontinuous phase transition with a nonzero gap r cr (18)while the dashed line is a continuous transition.Coming back to the range (1,2)of βone can determine the singularity type of the thermodynamical functional φ(κ,β)at the transition ie when κ→κcr .For small µ:g β(µ)=Γ(1−β)µβ−1+∞ k =0ζ(β−k )−µk−4.0−3.0−2.0−1.00.0 1.0 2.0κ−5.0−4.0−3.0−2.0−1.0−βCrumpledFluidFluid(Condensed)Figure 4:The phase diagram.The continuous line denotes a first-order phase transition.The dashed line denotes a continous transition.The dotted line shows the transition line that appears after addition of the artificial cut-off.For small µ∗we can write :κ−κcr =−log g β(µ∗)+log ζ(β)∼µβ−1∼r (β−1)/(2−β).(26)Inverting this with respect to r we eventually get :r =r ∗∼(κ−κcr )x ,where x =2−βanalysis can be generalised for an arbitrary set of weights.In general,the transition takes place when the argument e−µof the series(14)approaches the radius of convergence of the series.Denote the radius by e∆.In our example the radius was1ie∆=0,but in general∆may be anyfinite number.The condition for the existence of the transition is that the generating function K(14)isfinite at the radius of convergence of the series:K(∆,β)<∞.(28) The family of weights may be parametrised by many parameters.In the last formula,the parameters are represented by a single symbolβ.The equation:−κcr=K(∆,β),(29) is the equation of the critical line between the phases of the(κ,β)ensemble. Moreover,as one can see by repeating the arguments previously used for the particular weights(3),the condition for the transition to be discontinuous is:∂µK(∆,β)<∞(30) since the inverse of the derivative∂µK corresponds to the latent heat.The transition is continuous forβ’s for which the latent heat vanishes and condi-tion(28)is fulfilled.The position of the critical line on the phase diagram can be easily changed by a slight modification of weights.For example,a simple mul-tiplication by a constant C:p(q)→Cp(q)corresponds to a horizontal shift of the critical line in the phase diagram(Fig.4).In general a slight modifi-cation of weights can also change theβposition of critical line and thus the critical exponent(27).In this respect this model is“non-universal”(see also [9]).So far we have discussed the behaviour of the model in the thermodynamic limit.We now determine the exponents y(4)and Y(5).It is convenient to define the grand canonical partition function:Z(µ,κ,β)=∞N=1e−NµZ(N,κ,β)∼(µ−µ0)−1−Y,(31)whose singularity type depends on the exponent Y.The critical value of the chemical potential isµ0=φ(κ,β)is zero in the crumpled and nonzero in the fluid phase.The value of the exponent Y depends also on the position(κ,β) in the phase diagram.From(2)one obtains the following expression:Z(µ,κ,β)=eκgβ(µ)from which one can extract the singularity.In thefluid phase,the singularity comes from the residual denominator and it is:Z(µ,κ,β)∼(µ−µ0)−1,(33) from which one concludes that Y=0.For(κ,β)in the crumpled phase,the singularity comes from the numer-ator of the fraction(32)which itself has a power like singularity whenµapproachesµ0=0.ThusZ(µ,κ,β)∼µβ−1(34) and hence Y=−β.In other words,the partition function inherits the singularity from the weights or more precisely from the generating function K(µ,β).This effect comes from the dominance of the singular box contribu-tion over the negligible entropy.In general the singularity need not be power like.For example the weights:p(q)=exp(−aq+bq1/2),where a and b are positive constants,would lead to an essential singularity.In this case N Y would not be the dominant correction to the partition function.Let us now determine the singularity type on the critical line.The sin-gularity comes from the denominator in the expression(32).It changes at β=2.Forβ>2the expansion of the denominator in smallµbegins with the linear term.Because the numerator approaches a non-vanishing constant forµ→0,we have:Z(µ,κ,β)∼µ−1,(35) and hence Y=0as in thefluid phase.For1<β<2the expansion of the denominator begins with the singular termµβ−1(16),thusZ(µ,κ,β)∼µ1−β,(36) and Y=−2+β.So far we have tacitly set q min=1,which meant that each box contained at least one ball.As a result,the system with N balls could have M boxes at most.Hence the maximal value of r is1.If one instead chooses q min to be larger than one then the maximal r is:r max=1/q min.(37) In this case,the analysis of the model proceeds exactly as before,except that in place of1for the upper limit one sets r max=1/q min.In particular,the range of integration in equation(6)is now(0,r max)and the derivative∂r f10is infinite at r max.The average r approaches asymptotically r max when κ→−∞.So the only essential difference to the previous analysis is the range of r.In particular,the phase structure of the model is unchanged.We shall call r max the natural limit for r.The phase structure of the model can be made more complex by intro-ducing a slight modification.Let us impose a kinematic upper limit on r. We shall call this the artificial limit and denote it by r art.The artificial limit obviously has to lie below the natural one,r art<r max.Now the partition function becomes:N Y e Nφ(κ,β)=N y+1r artd re N(f(r,β)+rκ).(38)The main difference between the natural and the artificial limit is that the derivative∂r f is infinite at the former andfinite at the latter.In the case of a model with the natural limit,this limit is never reached by the average r , or at worst reached only asymptotically forκ→−∞,whereas the artificial limit can be reached forfiniteκwhen the term rκdominates over f(r,β) in the integrand(38).If wefixβand changeκfrom large positive values towards large negative,the system isfirst in the condensed phase r =0, then at the critical value r jumps to r cr and later smoothly rises until it reaches r art where it sticks(seefigure5).Such a situation is very similar to the previous standard one.There exists,however,the possibility of quite different behaviour.For β>β∗whereβ∗is defined by the equation:r art=ζ(β∗)N,(40)−4.0−2.00.0κ0.000.100.200.300.400.500.600.70r *Figure 5:The dependence r ∗on κfor various values of βin presence of the artificial cut-offwhich is the fraction of all balls which are in the singular box.The quantities r and s play the role of the order parameters.We have :crumpled : r =0, s =1,fluid : r >0, s =0,condensed : r >0, s >0.(41)The phase diagram is very similar to the one of simplicial gravity with the matter fields or the measure term [7,8].If one naively applies the mean field model to 4d simplicial gravity one would get the following constraint for the orders of vertices :q 1+q 2+...+q N 0=5N 4(42)where N 0is the total number of vertices,q i the order of vertex i ,and N 4is the number of 4simplices.We thus have the correspondence N 0↔M ,κ0↔−κand N 4↔N .The minimal order of a vertex is q min =5,so thenatural limit r max=1/5.This means that:N01We have y rather than Y as we are dealing with afixed number of boxes ensemble in the branched polymer analogy.To calculateγand hence Y,y in the condensed phase we will use a heuris-tic argument.We know that Y=y since the phase is again on the kinematic border(10),as for the crumpled phase,but y is now the exponent of the fixed r=r art ensemble.As before,in the particular case r=1/2we can take the value ofγ=2−βfrom the branched polymer model([9]).If we assume that this relation holds independently of r,and that y=γ−1as conjectured above for the crumpled phase,we obtain Y=y=1−βwhich differs by one from the crumpled phase value.We think that the main feature of the mean–field solution is that the exponentγproperly reflects an universal behaviour,ie it is independent on the location in the phase diagram as long as the system is in thefluid phase. This is more important in the context of simplicial gravity than the particular predictions for the exponentγcalculated for our choice of power law weights. Meanfield theories will,in any case,generically predict critical exponents incorrectly.In summary,we have seen that the meanfield model of simplicial quantum gravity can account for many of the features seen in simulations of the full model.With the correct choice of ensemble(a varying number of boxes)we reproduce thefirst order nature of the transition for forβ∈(2,∞),while for β∈(1,2]it is continuous,disappearing altogether atβ=1.We have also remarked that taking further account of the local geometrical constraints in the model can be implemented as a lowered limit on the region of integration in the saddle point equation,which we have denoted as the artificial upper limit.With the artificial limit in place we found the modified phase diagram offigure4,which possesses a new condensed phase.This is similar to the crinkled phase of full simplicial gravity with matterfields or the(apparently equivalent)measure term[7,8].AcknowledgementsWe are grateful to B.Petersson and G.Thorleifsson for discussion.This pa-per was partially supported by KBN grants2P03B04412,2P03B00814and by the Polish–British Joint Research Collaboration Programme under the project WAR/992/137.P.B was supported by the Alexander von Humboldt Foundation.References[1]P.Bialas,Z.Burda,B.Petersson,J.Tabaczek Nucl.Phys.B495(1997)463.[2]P.Bialas,Z.Burda,D.Johnston Nucl.Phys.B493(1997)505.[3]P.Bialas,Z.Burda,Phys.Lett.B416(1998)281.[4]S.Catterall,J.Kogut,R.Renken Phys.Lett.B416(1998)274.[5]S.Catterall,J.Kogut,R.Renken Phase structure of dynamical trian-gulation models in three-dimensions.hep-lat/9712011(To be published in NP)[6]Z.Burda,Acta Phys.Pol B29(1998)573.[7]S.Bilke,Z.Burda,A.Krzywicki,B.Petersson,J.Tabaczek,G.Thor-leifsson,Phys.Lett.B418(1998)266.[8]S.Bilke,Z.Burda,A.Krzywicki,B.Petersson,J.Tabaczek,G.Thorleif-sson,4-d simplicial quantum gravity:matterfields and the corresponding effective action,hep-lat/9804011to be published in Phys.Lett.B[9]P.Bialas,Z.Burda Phys.Lett.B384(1996)75.[10]J.Ambjorn,J.Jurkiewicz,Nucl.Phys.B451(1995)643.[11]D.Walkup,Acta Math.12575.。
