Houssem-Eddine Chihoub

Shadi Ibrahim

Gabriel Antoniu
email: gabriel.antoniu@inria.frmperez@fi.upm.es

María S Pérez

Consistency in the Cloud: When Money Does Matter!

With the emergence of cloud computing, many organizations have moved their data to the cloud in order to provide scalable, reliable and highly available services. To meet the ever-growing user needs, these services mainly rely on geographically-distributed data replication to guarantee good performance and high availability. However, with replication, consistency comes into question. Service providers in the cloud have the freedom to select the level of consistency according to the access patterns exhibited by the applications. Most optimizations efforts then concentrate on how to provide adequate trade-offs between consistency guarantees and performance. However, as the monetary cost completely relies on the service providers, in this paper we argue that monetary cost should be taken into consideration when evaluating or selecting a consistency level in the cloud. Accordingly, we define a new metric called consistency-cost efficiency. Based on this metric, we present a simple, yet efficient economical consistency model, called Bismar, that adaptively tunes the consistency level at runtime in order to reduce the monetary cost while simultaneously maintaining a low fraction of stale reads. Experimental evaluations with the Cassandra cloud storage on the Grid'5000 testbed show the validity of the metric and demonstrate the effectiveness of the proposed consistency model.

I. INTRODUCTION

Cloud computing has recently emerged as a popular paradigm for harnessing a large number of commodity machines. In this paradigm, users may rent computational and storage resources with respect to a pricing scheme similar to the economic exchanges in the utility market place: users can lease the resources they need in a Pay-As-You-Go manner. With data growing rapidly and applications becoming more data-intensive, many organizations have moved their data to the cloud aiming at providing scalable, reliable and highly available services. Cloud providers allow service providers to deploy and customize their environments in multiple physically separate datacenters to meet the ever-growing users' needs. Services therefore can replicate their state across geographically diverse sites and direct users to the closest or least loaded site. Replication has become an essential feature in storage systems and is extensively leveraged in cloud environments [START_REF] Ghemawat | The Google file system[END_REF] [START_REF] Ibrahim | Maestro: Replicaaware map scheduling for mapreduce[END_REF]. It stays behind several features such as fast access, enhanced performance, and high availability. For fast access: user requests can be directed to the closest datacenter in order to avoid communication delays and thus insure fast response time and low latency. For enhanced performance: in order to avoid overloading one single copy of the data, user requests can be re-directed to other replicas within the same datacenter (possibly different racks), to improve performance under heavy load. For high availability: failure and network partitions are common in large-scale distributed systems; by replicating, single points of failure can be avoided.

A particularly challenging issue that arises in the context of storage systems with geographically-distributed data replication is how to ensure a consistent state of all the replicas. Insuring strong consistency by means of synchronous replication introduces an important performance overhead due to the high latencies of networks across datacenters (the average round trip latency in Amazon sites varies from 0.3ms in the same site to 380ms in different sites [START_REF] Li | Making geo-replicated systems fast as possible, consistent when necessary[END_REF]). Consequently, many Internet services tend to rely on storage systems with eventual consistency. Eventual consistency allows the system to return some stale data at some points in time, but ensures that all data will eventually become consistent. Recently many cloud storage systems have been developed, such as Dynamo [START_REF] Decandia | Dynamo: amazon's highly available key-value store[END_REF], Cassandra [START_REF] Lakshman | Cassandra: a decentralized structured storage system[END_REF], BigTable [START_REF] Chang | Bigtable: A distributed storage system for structured data[END_REF], Yahoo! PNUTS [START_REF] Cooper | Pnuts: Yahoo!s hosted data serving platform[END_REF], and HBase [START_REF]Apache HBase[END_REF]. These solutions are practical to use as cloud and web service storage backends. They allow many web services to scale out their systems in an extreme way, while maintaining performance with very high availability. For example, Facebook uses Cassandra to scale out to host data for more than 800 million active users [START_REF]Facebook Statistcs[END_REF]. However, the undoubted availability and performance of such solutions prove to be too costly in terms of inconsistency ([10] reports the stale read rate was 66.61% under heavy access). Consistency-performance and consistency-availability trade-offs have long been investigated in literature: many consistency optimization solutions have been devoted to improving the application throughput and/or latency while preserving acceptable stale reads rate.

However, in the area of cloud computing, the economic cost of using the rented resources is very important and should be considered when choosing the consistency policy. To address these issues, this paper makes the following three contributions: [i] Service/bill details. To our knowledge, this is the first study to provide in-depth understanding of the monetary cost of cloud services with respect to their adopted consistency models. We discuss the different resources contributed to a service and the cost of these resources. This paper introduces an accurate decomposition of the total bill of the service into three parts with respect to the contributed resources: virtual machine (VM) instances cost, storage cost and network cost. To complement our analysis, a series of experiments are conducted to measure the monetary cost of different consistency levels in the Cassandra system [START_REF] Lakshman | Cassandra: a decentralized structured storage system[END_REF] on Grid'5000 [START_REF] Jégou | Grid'5000: a large scale and highly reconfigurable experimental grid testbed[END_REF] and Amazon EC2 [START_REF]Amazon Elastic Compute Cloud (Amazon EC2)[END_REF]. Such a study is important as a big-picture understanding of the consistency in geo-replicated systems must take into account the monetary cost within the cloud. [ii] Novel metric. We define a new metric called consistency-cost efficiency to evaluate consistency in the cloud. [iii] Equitable consistency and low cost. Based on our metric, we introduce a simple yet efficient approach named Bismar, which adaptively tunes the consistency level at runtime in order to reduce the monetary cost while simultaneously maintaining a low fraction of stale reads. Bismar relies on a consistency probabilistic model that estimates the stale reads rate and the relative costs of the application according to the current read/write rate and network latency.

We have implemented Bismar with intensive evaluations on the Cassandra cloud storage system on Grid'5000 [START_REF] Jégou | Grid'5000: a large scale and highly reconfigurable experimental grid testbed[END_REF]. We use the Yahoo! Cloud Serving Benchmark (YCSB) [START_REF]Yahoo Cloud Serving Benchmark[END_REF] to mimic a real cloud serving environment with elastic access pattern workloads. We show that Bismar can lead to efficient costs without exceeding the number of stale reads tolerated by the applications. Our paper is the first to provide a thorough analysis of the consistency cost in cloud storage systems. We view our work as a necessary step for bridging the gap between the business model of the cloud and the research community in distributed systems aiming at designing and building more efficient and more economical consistency models for cloud services.

II. RELATED WORK AND MOTIVATION

A. Related Work

Eventual consistency has been extensively exploited in literature and commercial products such as Dynamo [START_REF] Decandia | Dynamo: amazon's highly available key-value store[END_REF] in Amazon, Cassandra [START_REF] Lakshman | Cassandra: a decentralized structured storage system[END_REF] in Facebook and PNUTS [START_REF] Cooper | Pnuts: Yahoo!s hosted data serving platform[END_REF] in Yahoo!. Most of the work in literature have been dedicated to either measuring the actual provided consistency in cloud storage platforms [START_REF] Wada | Data consistency properties and the trade-offs in commercial cloud storage: the consumers' perspective[END_REF][14] [START_REF] Bermbach | Eventual consistency: How soon is eventual? an evaluation of amazon s3's consistency behavior[END_REF], or on adaptive consistency tuning in cloud storage systems [16][17] in order to meet the performance requirements of applications and reduce the consistency violation. In contrast, we focus on the monetary cost: we aim to provide an adaptive consistency approach that is cost efficient and does not violate the applications needs.

A closely related work on improving the monetary cost of consistency in the cloud is [START_REF] Kraska | Consistency rationing in the cloud: pay only when it matters[END_REF]. Kraska et al. [START_REF] Kraska | Consistency rationing in the cloud: pay only when it matters[END_REF] propose consistency rationing: an approach that adapts the level of consistency at runtime considering the monetary cost. The authors define consistency levels at data level (i.e., categorizes the data into three types and provides a different consistency treatment for each category). Consistency rationing at data level may incur additional metadata management overhead when the data size is large, our work therefore is at a transaction level: our adaptive tuning approach selects the number of replicas involved in an operation considering the best trade-off between consistency and monetary cost. The results discussed in our paper complement Kraska' work: monetary cost-oriented consistency approach at transaction levels to complement their work at data level.

B. Monetary Cost of Consistency: Why it does matter!

We observe that stronger consistency by the means of synchronous replications may introduce high latencies due to the cross-sites communication and therefore will significantly increase the monetary cost of the services: (1) High operation latency causes high monetary cost. Obviously because the cost of leasing a VM-instance is proportional to the latency (runtime). In addition to the increased cost of both the storage (e.g., number of IO requests) and the communication cost (e.g., number of cross-sites communication) due to the synchronous cross-site replication, (2) High operation latency causes significant financial losses for service providers that use such storage systems. For instance, the cost of a single hour of downtime for a system doing credit card sales authorizations has been estimated to be between 2.2M$-3.1M$ [START_REF] Anderson | What consistency does your key-value store actually provide?[END_REF]. On the other hand, we observe that eventual consistency or weaker consistency may reduce the monetary cost with respect to a lower maintained latency and therefore lower instance costs, but this comes at the risk of increasing the rate of stale data (e.g., [START_REF] Wada | Data consistency properties and the trade-offs in commercial cloud storage: the consumers' perspective[END_REF] demonstrated that under heavy access some of these systems may return up to 66.61% stale reads). This in turn adversarily impacts the financial profit of the service providers: it generates significant financial losses as it violates the SLAs of services users. This makes eventual consistency a two-edged sword. While the eventual consistency has been exploited extensively in literature and commercial products, its monetary cost and negative impacts on the stale reads rate have been largely ignored. The aforementioned observations, combined with the urgent need to address the consistencycost efficiency and stale reads problems associated with quorum replications, motivate us to an in-depth study of the monetary cost of the different consistency levels in the cloud and -as a result -to propose our cost efficient optimization.

III. MICROSCOPIC OF CONSISTENCY COST

In general, services require a set of linked servers (distributed in multi-sites) to run the applications; these servers are attached to a group of storage devices which store services data. With respect to cloud resources offers, a basic service bill will include charges for the following resources:

[i] Computing resources. Virtual machines equipped with a certain amount of CPU and memory resources and typically charged for the incurred virtual machine hours. [ii] Storage resources. Taking Amazon Web Services as an illustrating example, there are two representative storage services: Amazon Simple Storage Service (Amazon S3) and Amazon Elastic Block Store (Amazon EBS). The storage services are typically billed according to the used GBs per month and number of requests to the stored data. Taking into account the tremendous amount of data that current services need to manage and maintain, and the need to reduce the latency of data movement when processing data, Amazon EBS becomes the customer's first choice to achieve not only highly scalable and high performance services but highly reliable and predictable ones as well. This is despite the fact that Amazon EBS can be attached to any running Amazon EC2 instance and can be exposed as a device within the instance. Consequently, in this study we adapted the Amazon EBS pricing scheme. [iii] Network resources. The network cost is usually embedded within the cost of other services (computational service and storage services), and it varies in accordance to the service type and within/across sites.

Hereafter, we present a detailed analysis of the consistency cost in the cloud, using a widely used open source cloud storage system that supports multi-level consistency as an illustrated example, namely Cassandra [START_REF] Lakshman | Cassandra: a decentralized structured storage system[END_REF]. In Cassandra, the consistency level may be chosen on a per-operation basis and is represented by the number of replicas in the quorum (a subset of all the replicas). Ideally, we would like to get a deep idea of why different consistency levels may result in different costs, how the resources accordingly contribute to the total cost, and how background operations such as read repair can impact the overall cost. The choice of consistency level cl affects all of these three costs. When higher consistency levels are required, more replicas are involved in the requests. That affects both operations latency and throughput, which leads to a higher runtime. Similarly, network traffic grows higher with higher consistency levels, which leads to a higher networking bill. Moreover, higher consistency levels generate a higher number of requests from storage devices, directly affecting storage cost. Formula 1 presents the overall cost for services with geo-distributed replication for a given consistency level cl. Essentially, this cost is the combination of the VM instances cost Cost in (cl), the backend storage cost Cost st (cl), and network cost Cost tr (cl).

Cost all (cl) = Cost in (cl) + Costtr(cl) + Costst(cl) (1)

A. Computing unit: Instances cost

A common pricing scheme used by recent cloud providers is primarily based on virtual machine (VM) hours. Formula 2 presents the cost of leasing nbInstances VM-instances for a certain time (runtime).

Cost in (cl) = nbInstances × price × ⌈ runtime timeUnit ⌉ (2)
Here the price is the dollar cost per timeUnit1 (e.g., In Amazon EC2 small instance the price is $0.065 per hour).

In order to generalize our pricing model and avoid inaccurate pricing due to unexpected network behavior , we present the runtime in the form of number of operations nbOps in the workload while fixing the average throughput(ops/s) of a specific consistency level (runtime = nbOps throughput ). The throughput varies from one consistency level to another according to the size of the internal traffic between sites.

B. Storage cost

As mentioned earlier the storage cost includes the cost of leased storage volume (GBs per month) and the cost of I/O requests to/from this attached storage volume. In Amazon EC2 for instance, this would be the cost of attaching Amazon EBS to VM-instances in order to increase the storage capacity using a highly durable and reliable way. The total storage cost is accordingly given by Formula 3.

Costst(cl) = costPhysicalHosting + costIORequests (3)
Based on the size of hosted data (including all data replicas) nbNodes × dataSize where dataSize is the average data size per volume attached to VM-instance ( locality and load balancing are important features in current datacenters), we calculate the costPhysicalHosting in Formula 4.

costPhysicalHosting = nbNodes × ⌈ dataSize sizeUnit ⌉ × price (4)
where the price is the dollar cost per sizeUnit (e.g., In Amazon EBS the price is 0.10 per GBmonth).

We further estimate costIORequests in Formula 5.

costIORequests = cl × nbOps + readRepairIO nbRequestsUnit × price ( 5
)
where nbOps is the number of operations with respect to the consistency level cl (it varies according to the number of replicas involved in an operation). readRepairIO is the number of operations generated by the read repair operations. The read repair is a background operation that is mostly triggered when inconsistency is detected (more details about the read repair will be provided in Section III-C).

C. Network cost

Network cost varies in accordance to the service type of the source and destination and whether the data transfer is within or across sites. In general, inter-datacenter communications are more expensive than intra-datacenter communications. Formula 6 shows the total cost of network communications 2 as the sum of inter-datacenter (trafficInterDC and intra-datacenter trafficIntraDC communications cost.

Costtr (cl) = price(interDC) × ⌈ trafficInterDC sizeUnit ⌉ +price(intraDC) × ⌈ trafficIntraDC sizeUnit ⌉ (6)
Hereafter, we illustrate how to estimate both the inter-and intra-datacenter traffic. Formula 7 shows our model of the inter datacenter, trafficInterDC, given the replicas communication (interDcRep), the request routing (request routing), and the internal mechanisms traffic (IMechTraffic). trafficInterDC = interDcRep + requestRouting + IMechTraffic [START_REF] Cooper | Pnuts: Yahoo!s hosted data serving platform[END_REF] The inter-site traffic generated by the replicas communications strongly depends on the cl and the distribution of replicas among datacenters (i.e., the number of replicas involved in a request to other datacenters which can be estimated as ⌊(nbDc -1)× cl nbDc ⌋ 3 where nbDc is the number of datacenters). Formula 8 shows our estimation of the inter traffic generated by the replicas communications. [START_REF]Apache HBase[END_REF] where avgDataSize is the average data size needed to be propagated to other replicas for one operation.

InterDcRep = ⌊(nbDc -1) × cl nbDc ⌋ × AvgDataSize × nbOps
The traffic generated by the request routing and internal mechanisms depends essentially on the storage system design and implementation. Since our approach is destined to run on Cassandra storage, hereafter we illustrate such values with respect to this particular storage system. In Cassandra, all nodes (peers) have equal ranges of data and thus have an equal number of keys: this implies that each node is responsible for 1 number of nodes fraction of the keys. Giving the number of nodes as nbNodes and the average number of nodes per datacenter avgNodesDc, the average number of request routing for an operation can be estimated as nbNodes-avgNodesDc nbNodes . The size of inter traffic generated by request routing for a number of operations nbOps is therefore denoted as Formula 9.

requestRouting(interDC) = nbNodes -avgNodesDc nbNodes ×nbOps × avgDataSize [START_REF]Facebook Statistcs[END_REF] In Cassandra, the main internal traffic is generated by the gossip traffic and the read repair as shown in Formula 10. IMechTraffic = gossip(interDc) + readRepair(interDc) [START_REF] Wada | Data consistency properties and the trade-offs in commercial cloud storage: the consumers' perspective[END_REF] The gossip traffic -used to share the state of nodes in the ring -is relatively small since it is just transmitting the state of one node which is negligible compared to data transfer. On the other hand, the read repair is used to propagate data to out-of-date (stale) replicas. The read repair is triggered in two cases: (1) At random times for some requests: defined 3 For example if the (nbDC=3) and number of replicas involved in an operation (cl=4), the estimated number of replicas involved in a request on other dataceters is ⌊2 × 43 ⌋ = ⌊ 8 3 ⌋ = 2 where ⌊ ⌋ is a floor function.

by the system administrator; (2) Whenever inconsistency is detected. Formula 11 shows that read repair traffic depends on the probability or chance of triggering the mechanism rrChance which is defined by the storage administrator, as well as the chance of detecting mismatching replicas' timestamps mmChance = rf-cl rf × nbWrites nbReads+nbWrites , where rf is the replication factor, nbWrites and nbReads are the number of writes and reads.

readRepair(interDC) = nbOps × avgDataSize ×(rrChance × ⌊ rf nbDc ⌋ + mmChance × ⌊ rf -cl nbDc ⌋) (11)
Computing the intra datacenter traffic size is very similar to the one of inter datacenter traffic. Due to the page limitation we didn't include its analysis here (Please refer to our technical report [START_REF] Chihoub | Consistency in the cloud: When money does matter![END_REF] for more details).

D. Practical View of Consistency Cost in Cassandra

We complement and benefit from our earlier analyze, by evaluating the monetary cost in Cassandra. Experimental setup. We run our experiments on Grid'5000 [START_REF] Jégou | Grid'5000: a large scale and highly reconfigurable experimental grid testbed[END_REF] and Amazon Elastic Compute Cloud (EC2) [START_REF]Amazon Elastic Compute Cloud (Amazon EC2)[END_REF]. On Grid'5000, we deployed Cassandra on two datacenters (sites): with 30 nodes on the Sophia site and 20 nodes on the Nancy site as shown in Figure 1(a). All the nodes in Sophia are equipped with a 250GB hard disk, 4GB of Memory, and 4-cores AMD Opteron. The nodes in Nancy are equipped with disks of 320GB space, 16GB of Memory, and 8-cores Intel Xeon. The network connection between the two sites is provided by RENATER (The French national telecommunication network for technology, education, and research). It consists of a standard architecture of 10Gbit/s dark fibers. The network route between the two sites is the following: Nancy-Paris-Lyon-Marseille-Sophia: the average round trip latency is on average 0.230ms within the same site and 18.2ms in-between the two sites. On Amazon EC2, we also deployed Cassandra on 18 large instances (m1.large) on two availability zones: 10 instances on us-east-1a and 8 instances on us-east-1d. The average round trip latency is on average 0.284ms within the same site and 0.813ms inbetween the two availability zones.

We used Cassandra-1.0.2 with a replication factor of 5 replicas: 2 replicas are allocated in Nancy and 3 replicas in Sophia (The same replication factor is used in Amazon EC2: 2 replicas in us-east-1d and 3 replicas in us-east-1a). Our replication strategy uses NetworkTopologyStrategy to enforce replication across multiple datacenters. We adopt the pricing schemes from Amazon web services as shown in Table 1 4 . We study the cost variation by evaluating different consistency levels (e.g., eventual consistency: ONE, TWO, Quorum: THREE, and strong consistency: ALL). Micro Benchmark. We aim at a micro benchmark representing typical workloads in current services hosted in clouds. Based on case studies [START_REF] Cooper | Pnuts: Yahoo!s hosted data serving platform[END_REF][20], we have selected the Yahoo! Cloud Serving Benchmark (YCSB) [START_REF]Yahoo Cloud Serving Benchmark[END_REF]. YCSB is used to benchmark Yahoo! storage system "PNUTS" [START_REF] Cooper | Pnuts: Yahoo!s hosted data serving platform[END_REF]. It is extended to be used with a variety of open-source data stores such as mongoDB [START_REF]mongoDB[END_REF] and Cassandra [START_REF] Lakshman | Cassandra: a decentralized structured storage system[END_REF]. YCSB provides the features of a real cloud serving environment such as scale-out, elasticity and high availability. We use YCSB-0.1.3 and we run WorkloadA which is a heavy read-update workload (read/update ratio: 60/40). In both environments, our workload consists of 10 million operations on 5 million rows with a total of 23.84GB of data after replication.

Results. As shown in Figure 1(b), the total monetary cost decreases when degrading the consistency level: the cost reduces from $138.76 -when the consistency level is set to ALL -to $71.72 when the consistency level is ONE (i.e., weak consistency reduces the cost by almost 48%). This result was expected as lower consistency level involves fewer replicas in the operations, and thus maintaining low latency, less I/O requests to the storage devices, and less network traffic in general (the runtime of WorkloadA varies from 4 hours to 7 hours according to the consistency level). This cost reduction, however, comes at the cost of a significant increase in the stale reads rate: as shown in Figure 1(b) 79% of the reads are stale reads -only 21% of the reads are fresh reads -when the consistency level is set to ONE. Furthermore, it is obvious that degrading the consistency level to Quorum (here the number of replicas involved in an operation is 3 replicas) reduces the total cost by 13% while maintaining a zero stale reads rate as shown in Figure 1(b). This is because the storage system intends to answer the read requests with the most up-to-date replicas (fresh reads). Moreover, degrading the consistency level to TWO reduces the total monetary cost by almost 36%, but it adversary impacts the system consistency: only 61% of the reads where fresh reads. Figure 1(c) shows the breakdown of the total cost according to the contributed resources. In general, the instances cost has the higher cost amongst other resources (storage and network): it contributes to almost 90% of the service bill while the storage and network contribute on average to only 9% and 0.4%, respectively. This is due to our experiments' scale -number of operations -and the cheap prices of resources (as shown in Table I the intra communication is free of charges). As shown in Figure 1(c), storage cost has a relatively lower contribution to the total cost for stronger consistency (ALL and Quorum) compared to weaker consistency (ONE and TWO): it contributes on average to 7.2% for the stronger one and 9% for the smaller one. The ALL consistency level requires higher nbOps compared to Quorum while both have zero/low readRepairIO and thus according to Formula (6) ALL has a relatively higher storage cost contribution in contrast to Quorum (e.g., it is 7% for Quorum and 7.5% for ALL). Moreover, although the nbOps is smaller for (ONE and TWO) compared to (ALL and Quorum) but the increasing number of readRepairIO increases the storage cost. Furthermore, as the cost of readRepairIO is proportional to the rate of stale reads, ONE has higher storage cost contribution in contrast to TWO. Network cost has also relatively a lower contribution to the total cost for stronger consistency (ALL and Quorum) compared to weaker consistency (ONE and TWO): it contributes on average to 0.175% for the stronger one and 0.275% for the smaller one. The ALL consistency level requires higher interDcRep compared to Quorum (higher number of involved replicas as well as Quorum always tends to answer the requests by involving the most close replicas "within the same datacenter if possible") while both have zero/low IMechTraffic and thus according to Formula 10 ALL has a relatively higher network cost contribution in contrast to Quorum. Moreover, although the interDcRep is smaller for (ONE and TWO) compared to (ALL and Quorum) but the increasing size of IMechTraffic -due to the high rate of stale reads -increases the network cost. Amazon EC2: Figures 1(d) and 1(e) support our earlier findings and observations with Grid'5000 (for more details please refer to [START_REF] Chihoub | Consistency in the cloud: When money does matter![END_REF]). The total cost variation in Amazon is lower than in Grid'5000, because of the more powerful machines and the lower cross-sites latency. Moreover, the costs of the ONE and TWO levels are the same, although there were significant variations in the running time (2hours and 1minute for ONE and 2hours and 33minutes for TWO) and also significant variations in the network traffic and storage requests. This is because of the coarse-grained pricing units (hour and GBs , etc).

IV. CONSISTENCY-COST EFFICIENCY METRIC

As discussed in the previous sections, data consistency can strongly impact the financial cost of a given service (i.e., while stronger consistency with high latency implies higher monetary cost of an operation, the weaker consistency with high rate of stale rate causes financial loss). Consequently, monetary cost should be considered when evaluating the consistency in the cloud [START_REF] Kraska | Consistency rationing in the cloud: pay only when it matters[END_REF]. As cloud computing is an economy-driven distributed system where monetary cost is explicit and measurable metric [START_REF] Wang | Distributed systems meet economics: pricing in the cloud[END_REF][23], we argue that the consistency-cost trade-off can be easily exposed in the cloud. Therefore in this paper, we define a new metricconsistency-cost efficiency -that exposes the tight relation between the degree of achieved consistency for a given monetary cost. Our goal is to define a general yet accurate metric to evaluate consistency and thus using this metric as an optimization metric for cloud systems. Accordingly we define the consistency-cost efficiency as the ratio of consistency, measured by the rate of fresh reads, to the relative consistency cost as shown in Formula 12.

Consistency-Cost Efficiency

= Consistency(cl) Cost rel (cl) (12)
Where Consistency(cl) = 1-stale reads rate and Cost rel is the relative consistency cost with respect to the strong consistency and given as Cost rel (cl) = Cost(cl) Cost(cl all) . It is important to mention that our metric is designed and can only be applied when strong consistency is not required by an application: we can consider our metric as a system optimization for eventual consistency (i.e., tune the consistency to reduce the monetary cost without violating the application's requirements of fresh read rate).

V. BISMAR: ECONOMICAL CONSISTENCY APPROACH

We design and implement our approach with the following goals: [A] Extendable consistency-cost efficiency.

Our solution aims at providing consistency guarantees while reducing the monetary cost. Therefore, we propose to use the consistency-cost efficiency (described in section IV) as an optimization metric: simply by selecting the consistency level with maximum consistency-cost efficiency. Moreover, to meet the diversity of applications requirements (e.g., cost constraint and fresh reads rate constraint), our solution can be easily extended to enable consistency-cost efficiency while favoring either cost or consistency. [B] Selfadaptive. With the ever growing diversity in the access patterns of cloud applications along with the unpredictable diurnal/monthly changes in services loads, it is important to provide a self-adaptive approach that transparently scales the consistency level up/down at runtime without any human interaction. Our approach, therefore, embraces an estimation model for consistency-cost efficiency that could be achieved with different consistency levels: at runtime, the application's access pattern and network latency are fed to the consistency probabilistic estimation model (we have extended the model in [START_REF] Chihoub | Harmony: Towards automated self-adaptive consistency in cloud storage[END_REF] as will be explained later in this section) in order to estimate the rate of stale data that could be read in the storage system. Furthermore, we use the same information (e.g., access pattern and network latency) along with the predicted stale read rate (i.e., to estimate the number of stale reads) to compute the monetary cost. [C] Pricing independent. Our solution targets public cloud and is not limited to any cloud provider in terms of provided services or pricing schemes. The fine-grained monetary cost analysis that is used for cost estimation (introduced in Section III) can be easily adopted to different services and pricing. [D] Cloud storage systems independent. Since our solution is implemented as a separate layer at the top of the cloud storage system, it does not impose any modifications to the cloud system code. Our approach, therefore, can be applied to different cloud storage systems that are featured with flexible consistency rules.

Consistency Probabilistic Estimation.

In our previous work [START_REF] Chihoub | Harmony: Towards automated self-adaptive consistency in cloud storage[END_REF], we proposed an estimation of the stale reads rate in the system by means of probabilistic computations. This estimation model requires basic knowledge of the application access pattern and of the storage system network latency. Network latency in this case is of high importance, since it is the determinant of the updates propagation time to other replicas. The access pattern, which includes read rates and write rates is a key factor to determine consistency requirements in the storage system.

Hereafter, we briefly describe our extended estimation model (for more details readers can refer to [START_REF] Chihoub | Harmony: Towards automated self-adaptive consistency in cloud storage[END_REF]). Essentially, the client gets stale data when a read operation is performed from one or more replicas that have not been updated yet, while an update process is going on. Transactions arrivals are almost exclusively considered as Poisson process as it is a common way to model them [START_REF] Chihoub | Harmony: Towards automated self-adaptive consistency in cloud storage[END_REF]: we assume that the writes and the reads arrivals follow the Poisson distribution of parameter λ -1 w (we chose λ -1 w instead of λ w in order to simplify subsequent formulas where the parameter will be inverted) and λ r respectively. Since the distribution of waiting time between two Poisson arrivals is an exponential process. We assume the stochastic variables X w and X r of write time date and read time date follow exponential distributions of parameters λ -1 w and λ r respectively.

The probability of a stale read P r(staleRd), assuming that writes are performed with a consistency level that involves only one replica, is given by Formula 13 where rf is the replication factor, and T p is the average time to propagate an update to other replicas. P r(staleRd) = (rf -1)(1 -e -λr Tp )(1 + λr λw) rf λrλw [START_REF]Yahoo Cloud Serving Benchmark[END_REF] Given that when the storage system supports multiple consistency levels, the consistency level for read and write operations (cl r and cl w respectively) may vary with time. Accordingly, we extend the probability model in Formula 13 to consider all the consistency levels for write and read operations that are smaller or equal to the Quorum level, where a Quorum is computed as: ⌊ replicationf actor 2 + 1⌋. This probability is given in Formula 14.

Efficiency-aware algorithm. Many applications do not strictly require strong consistency: a consistency optimization solution, therefore, can be introduced to improve system throughput, latency and monetary cost. To achieve this goal we consider our metric as an optimization metric as shown in the following algorithm. At runtime, our system feeds the efficiency-aware algorithm with data related to the system read/write rates along with the network latency. These data are used by the consistency probabilistic estimation model to compute the expected achieved fresh reads when using different consistency levels. The relative monetary cost is also computed according to the system configuration and the stale read estimation. So the algorithm selects the consistency level that offers the most equitable consistency, cost trade-off (the maximum consistency-cost efficiency value).

VI. EVALUATION

We have built our approach as a separate layer at the top of Apache Cassandra-1.0.2 [START_REF] Lakshman | Cassandra: a decentralized structured storage system[END_REF]. The core of this layer consists of two modules. Both modules were implemented in Python 2.7. The monitoring module collects relevant metrics (data) needed for our approach of the storage system's information. The data is further communicated to the dynamic consistency module. An estimation of consistency-cost efficiency is computed -according to the estimated stale reads rate and the monetary cost (instance, storage and network cost)and then compared in order to provide a cost efficient consistency level for the running application at that point of time. Later in this section, we present our detailed evaluation of our consistency-cost efficiency metric and the Bismar prototype using Cassandra on Grid'5000 testbed. We also use YCSB to run WorkloadA (We used the same testbed and WorkloadA described in Section III-D). In order to present the dynamicity of the system (i.e., the variation of throughput and the read/write rates during the runtime), we ran the workload, varying the number of threads starting with 1 thread, then, 50, 20, 7 and finally, 30 threads. Effectiveness of the Consistency-cost efficiency. In order to validate our metric, we collect samples when running the same workload (with different consistency levels), varying the number of client threads, and thus exhibiting different access patterns. Figure 2(a) shows the results where each point represents a different access pattern. Higher consistency-cost efficiency values are associated with high rates of fresh reads (around 80%). This indicates the effectiveness of our metric: it is designed to achieve the best price without violating the consistency (we consider the 80% fresh reads as acceptable consistency). Monetary Cost and Performance improvement. Figure 2(b) shows the monetary costs of running the workload with the three static consistencies (ONE, TWO and Quorum) and with our dynamic adaptive approach. As expected, ONE exhibits the lowest monetary cost but at the cost of fresh reads. Our experiments also show some interesting results: Bismar achieves lower cost in contrast to the consistency level TWO. Since Bismar always selects the consistency level with the highest consistency-cost efficiency to adopt to the workload dynamicity, Bismar adopts the consistency level ONE for almost 70% of its running time while it adopts the consistency level Quorum for 30% of its running time as shown in Figure 2(c). As a result, the cost reduction when running with ONE overcomes the cost increase when running with Quorum. Since Bismar targets applications that do not require strong consistency, we consider Bismar as an eventual consistency optimization for cloud computing. Therefore, improves the monetary cost of services while maintaining acceptable rate of fresh reads. Accordingly, we compare the cost reduction and performance improvement by Bismar in contrast to the Quorum consistency level. As shown in Figure 2(b), Bismar reduces the monetary cost by almost 31.5% in contrast to Quorum level (From $456 to $312). The cost reduction is mainly due to the performance improvements (Bismar improves the overall response time by almost 32.2%). Staleness evaluation. Figure 2(d) shows the stale reads rates caused by different consistency approaches. It is clear that static levels ONE and TWO produce higher stale reads rate: 61% of the reads where on stale data with ONE and 36% with TWO. Moreover, the Quorum consistency level returns always up-to-date data (i.e., stale reads rate is 0%) because at least one replica with the freshest data should be in the Quorum. Bismar however, returns very small portion of stale reads (only 3%), but with very important money saving (31.55% cost reduction compared to Quorum). The 3% stale reads is considerably reasonable for many applications. Zoom on resources cost in Bismar. Figure 2(e) shows the breakdown of the total cost according to the contributed resources for different consistency levels and Bismar. As shown and discussed earlier in Section III-D, the instance portion of the total cost increases with upgrading consistency while the portion of both the storage and network costs increase with degrading the consistency level. However, the aforementioned observation is also applied on Bismar: comparing Bismar against Quorum, we notice that instance cost portion in Bismar is lower than in Quorum. Furthermore, we observe that the portion of both the storage and network costs in Bismar is higher than in Quorum. This can explain why the cost reduction was only 31.5% while the performance improvement was 32.2%: because of the adversary impacts of the storage and network costs in Bismar. Moreover, we observe that the portion of both the storage and network costs in Bismar is higher than in all static consistencies, because Bismar combines both the high number of requests when adopting a higher consistency level and also read repair cost when stale reads is detected when Bismar adopts lower consistency level.

VII. CONCLUSION

In this study, we investigate the monetary cost of consistency in the cloud. Our detailed analysis and study revealed a noticeable monetary cost variation when different consistency levels are used. As a first step to understand the impacts of the different consistencies on the monetary cost and fresh reads in the cloud, we define the consistencycost efficiency metric. Based on our metric, we introduce a simple, yet efficient approach, named Bismar, that adaptively tunes the consistency level at run-time in order to reduce the monetary cost while simultaneously maintaining a low fraction of stale reads. Bismar relies on a consistency probabilistic model that estimates the stale reads and the relative costs of the application according to the current read/write rate and network latency. We have implemented Bismar with extensive evaluations on the Cassandra cloud storage system using the Grid'5000 testbed. We show that Bismar can lead to efficient cost without exceeding the application tolerated number of stale reads. As future work, we intend to perform more detailed theoretical and empirical analysis of the consistency-cost efficiency metric with a broader set of application workloads. Also we are interested in building an efficient mechanism for dynamic resource provisioning based on our cost function.

Figure 1 .

 1 Figure 1. Experiments setup and results on Grid'5000 and Amazon EC2

  P r(staleRd) = (rf -(clw + clr -1))(1 -e -λr Tp )(1 + λr λw) rf λr λw

Figure 2 .

 2 Figure 2. (a) and (c) consistency-cost efficiency CCE-effectiveness and -distribution; (b), (d), and (e) Bismar evaluation on Gird'5000

We use the ceiling function because most providers charge each partial instance-hour as a full hour.

For simplicity, we consider only two geographical areas within which the prices differ. Some cloud providers may have more geographicallyoriented prices: within availably zone, within regions, between regions. However, our pricing model can be easily extended to any number of geographical-oriented pricing options.

The price of Amazon EC2 large instance was $0.32 at the time of writing this paper and it is now $0.26. However, as this price is applied to all consistency levels the difference in the pricing therefore doesn't affect our results and findings.

ACKNOWLEDGMENTS

This work is supported by the EU FP7 MCITN SCALUS project under grant agreement no. 238808, the Héméra INRIA Large Wingspan-Project, and the ANR MapReduce grant (ANR-10-SEGI-001). The experiments presented in this paper were carried out using the Grid'5000/ALADDIN-G5K experimental testbed.


F Gulminelli
email: gulminelli@lpccaen.in2p3.fr

Ad R Raduta

J Margueron

P Papakonstantinou

M Oertel

Some aspects of the phase diagram of nuclear matter relevant to compact stars

Dense matter as it can be found in core-collapse supernovae and neutron stars is expected to exhibit different phase transitions which impact the matter composition and the equation of state, with important consequences on the dynamics of core-collapse supernova explosion and on the structure of neutron stars. In this paper we will address the specific phenomenology of two of such transitions, namely the crust-core solid-liquid transition at subsaturation density, and the possible strange transition at super-saturation density in the presence of hyperonic degrees of freedom. Concerning the neutron star crust-core phase transition at zero and finite temperature, it will be shown that, as a consequence of the presence of long-range Coulomb interactions, a clusterized phase is expected which is not accessible in the grandcanonical ensemble. A specific quasi-particle model will be introduced and some quantitative results relevant for the supernova dynamics will be shown. The opening of hyperonic degrees of freedom at higher densities corresponding to the neutron stars core also modifies the equation of state. The general characteristics and order of phase transitions in this regime will be analyzed in the framework of a self-consistent mean-field approach.

Introduction

Triggered by the first data taking of X and gamma satellites, as well as by the improved capabilities of radio telescopes and high and ultra-high gamma ray detectors, there has been in the last ten years an impressive accumulation of observational data on supernova and neutron stars in different environments and at different evolution stages. Among the most important recent results one should mention the precise measurement of a very massive two solar-mass neutron star [1] which challenges a number of theoretical equations of state of dense matter, and the measurement of the cooling curve of the young pulsar Cassiopeia A [2], which gives strong constraints on the super-fluid properties of neutron star cores.

Understanding these data requires a good modelization of the supernova dynamics and of the formation process of neutron stars [3,4], but also a precise and detailed control of the relevant microphysics. Sophisticated two and three-dimensional core-collapse supernova simulations show that matter in the supernova core explores a very large interval of baryonic densities, ranging from about ρ > 10 10 g • cm -3 ≈ 10 -4 ρ 0 to several times the normal nuclear density ρ 0 , temperatures between some hundreds of keV and around 80 MeV, and proton fractions between ≈ 0.5 and ≈ 0.1.

Both the equation of state and the neutrino cross sections are strongly influenced by the possible presence of phase transitions in these wide thermodynamic conditions. In particular, first-order phase transitions lead to a softening of the baryonic pressure, while second order phase transitions lead to a dramatic increase of the electroweak interaction probability [5]. This is why a control of the phase diagram of stellar matter is of great importance in our understanding of these complex astrophysical process.

In this contribution, we will limit ourselves to the thermodynamic conditions where matter can be described by nucleonic or hadronic degrees of freedom and will examine two phase transitions which are expected to play a role in supernova physics, namely the crust-core phase transition at sub-saturation density and the possible phase transition towards strange matter at super-saturation density.

2. Sub-saturation densities and the quenching of the first-order phase transition Since the early days of the equation of state modelization it was clearly recognized that, because of the strong similarities between the effective nucleon-nucleon potential and molecular interactions, the phase diagram of diluted nuclear matter should present a first-order phase transition of the liquid-gas type, terminating at high temperature and density in a critical point [6]. This conjecture has been confirmed in all phenomenological [7,8] or microscopic [9] modelizations of symmetric and asymmetric nuclear matter with realistic effective interactions compatible with recent nuclear data.

In these calculations, nuclear matter is defined as an idealized bulk medium of neutrons and protons where the electromagnetic interaction is artificially switched off in order to achieve a thermodynamic limit. In the physical situation of stellar matter, however, charge neutrality is achieved by the screening effect of electrons on the proton charge, , which induces Coulomb effects that drastically modify the liquid-gas phase transition associated to uncharged nuclear matter. Because of the very different mass between electrons and protons, the compressibility of electron and proton matter is very different. Therefore, in all density and temperature conditions relevant for neutron star physics, the electron charge can be safely considered as uniformly distributed [10]. This means that any baryonic density fluctuation (which is correlated to a proton density fluctuation because of the repulsive symmetry energy) induces a fluctuation in the electric charge. A well known consequence of the resulting Coulomb correlations is that the low density phase at zero temperature is not a gas, but a Wigner crystal of nuclei immersed in the homogeneous electron background [11]. At finite temperature, the presence of the electron background leads to a quenching of the first order liqui-gas transition, which is replaced by a continuous transition through a clusterised medium [12]. This Coulomb frustration effect has been confirmed by different microscopic models [13] . A continuous evolution from the Coulomb lattice to a homogeneous nuclear fluid, passing through the formation of clusters of different sizes and strongly deformed dis-homogeneous structures close to the saturation density has been reported, both at zero and at finite temperature.

These calculations are numerically very heavy and a complete thermodynamic characterization of stellar matter under the frustrated phase transition in a microscopic model has not been done yet. Such a task can however be performed in the phenomenological so-called nuclear statistical equilibrium (NSE) approaches, which treat the bound states of nucleons as new species of quasi-particles [START_REF] Phillips | The Physics of Stars[END_REF][START_REF] Botvina | [END_REF]16,17].

A phenomenological model of dilute stellar matter

Within NSE, the baryonic component of the stellar matter is regarded as a statistical equilibrium of neutrons and protons, the electric charge of the latter being screened by a homogeneous electron background. As a first approximation, one can consider that the system of interacting nucleons is equivalent to a system of non-interacting clusters, with the nuclear interaction being completely exhausted by clusterization. This simple model can describe only diluted matter at ρ ρ 0 as it can be found in the outer crust of neutron stars, while nuclear interaction among nucleons and clusters has to be included for applications at higher density,when the average interparticle distance becomes comparable to the range of the force. A simple possibility [18,19] is to take into account interactions among composite clusters in the simplified form of a hardsphere excluded volume, and interactions among nucleons in the self-consistent Hartree-Fock approximation with a phenomenological realistic energy functional fitted on a large number of measured mass and radii. In this version of the model, the thermodynamics can be calculated in the ensemble (β, µ I , ρ) defined by inverse temperature β = T -1 , isovector chemical potential µ I = µ n -µ p and baryonic density ρ = ρ n + ρ p . The total entropy density is given by

σ can β,µ I (ρ) = ln Z a=1 β,µ I (ρ f ) + lim V →∞ 1 V ln Z a>1 β,µ I (V ρ cl ) (1)
where the density repartition between the clustered ρ cl and unbound ρ f component ρ = ρ cl + ρ f is uniquely defined by the condition of having a single chemical potential µ for both components. The unbound nucleons partition sum is calculated in the mean-field approximation:

ln Z a=1 β,µ I (ρ) = ln Z 0 β,µ,µ I -β ∂ ∂β ln Z 0 β,µ,µ I + V HF -βµρV, (2)
In this equation HF is the Hartree-Fock energy and the non-interacting part of the partition sum can be expressed as a functional of the kinetic energy density τ q for neutrons (q) or protons (q = p) :

ln Z 0 β,µ,µ I = 2βV 3 q,p τ q ∂ HF ∂τ q , (3)
where τ q are self-consistently derived as a function of the associate chemical potential µ q and effective mass m * q . The partition sum of clusters for a total particle number A = V ρ cl is calculated via a recursive relation:

Z β,µ I (A) = 1 A A a=2 aω a Z β,µ I (A -a). (4)
where the weight of the different cluster size is given by:

ω a = V F a i=-a 2πam 0 βh 2 3/2 exp -βf β a,i exp(βiµ I ). (5)
Here, f β a,i is a phenomenological free energy functional including the screening effect of electrons in the Wigner-Seitz approximation, and V F = V (1 -ρ cl /ρ 0 ) is the volume fraction available to the clusters [18,19] .

The thermodynamics of the extended NSE model was studied in ref. [19]. As it is wellknown when studying phase transitions with long range interactions [20], the different statistical ensembles are not equivalent and it is very important to use the modelization adapted to the physical situation under study. In the case of compact stars, the density profile is imposed by the gravitational field and the correct ensemble is the canonical one.

The phase transition quenching and the effect of the statistical ensemble is illustrated in the left part of Figure 1, which shows the cluster distributions at a given temperature. As a function of the density, the average cluster size is continuously changing and can never be assimilated to a portion of the fluid phase. The right part of the same figure compares the grandcanonical and canonical cluster distribution in a thermodynamic situation relevant for supernova physics. We can see that accounting for the correct thermodynamics has important consequences on the matter composition. Since the cluster abundances determine the electron capture rate, which in turn is a capital ingredient for the size of the homologous core and the cooling process, it is clear that a control of the phase diagram is important for supernova physics. It is also interesting to remark that the most probable abundances concern not so neutron-rich nuclei which are potentially accessible in laboratory experiments: a detailed knowledge of the mass, level density and electron capture cross section for these nuclei is thus needed to have a reliable description of the equation of state.

In-medium effects

In the extended NSE model presented in the previous section clusters are treated as an ideal gas characterized by an empirical energy functional which is modified with respect to nuclei in the vacuum only through the excluded volume mechanism. This is an improvement compared to standard NSE [START_REF] Phillips | The Physics of Stars[END_REF][START_REF] Botvina | [END_REF]16,17] in which the Coulomb screening by the electron background is the only interaction accounted for, but still represents a crude approximation of the effect of the medium [21]. Microscopic modifications of the cluster binding energies due to Pauli blocking shifts have been computed [9] and shown to correctly describe the Mott dissolution of deuterons in dense matter, but this heavy numerical approach cannot be extended to massive clusters. In the original Lattimer-Swesty model [22] the assumption is made that the effect of the medium consists in a modification of the cluster surface tension, the bulk energy being unaffected. However some interpretations of multifragmentation data suggest that bulk terms might also be affected [23].

To microscopically account for in-medium modifications of the cluster binding energies, we have realized a systematic set of Hartree-Fock calculations in the Wigner-Seitz cell for various isotopic chains with a total neutron number ranging from the stability line up to N ≈ 3000. The Left part: isoscalar and isovector bulk and isoscalar surface term of the in-medium modified mass formula extracted from Hartree-Fock calculations in the Wigner-Seitz cell. Right part: comparison between the microscopic Hartree-Fock volume energy per particle and surface energy per A 2/3 and the parameterization eq.( 6) for three isotopic chains as a function of the bulk proton-neutron asymmetry. The model surface term is given by a m S (ρ) -a m Ssym (ρ, δ)δ 0 2 , see eq. ( 6).

resulting microscopic nuclear energy, consisting of a volume and a surface term, is parameterized with an in-medium modified mass formula

E HF = HF (ρ n , ρ p )(V W S -V cl )-a m V (ρ) -a m V sym (ρ, δ)δ 2 0 A+ a m S (ρ) -a m Ssym (ρ, δ)δ 0 2 A 2/3 (6)
where V W S is the volume of the calculation cell and ρ n , ρ p , ρ, δ is the neutron, proton, total density and asymmetry δ = (ρ n -ρ p )/ρ of the free nucleons constituting the dripping gas. The cluster mass and charge A, Z and the gas densities are extracted through a Wood-Saxon fit of the local density. The bulk density ρ 0 and asymmetry δ 0 = (ρ 0n -ρ 0p )/ρ 0 in the model are approximated by analytical functions of A, Z, while V cl = A/ρ 0 . The resulting in-medium parameters obtained from a preliminary fit of the microscopic Hartree-Fock results through eq.( 6) are shown in the left part of Figure 2, while the quality of the fit is displayed in the right part of the same figure.

To obtain the fit shown in Fig. 2 we have employed the ansatz

a m S (ρ) = a S 1 - ρ ρ 0 γ 1 ; a m Ssym (ρ, δ) = a Ssym 1 - ρδ ρ 0 δ 0 γ 2 (7)
with γ 1 = γ 2 = 2 and with a Ssym = -32 MeV. A dependence of a m Ssym on ρ 0 ,δ 0 is implicit. We can see that the in-medium modified liquid drop expression eq.( 6) provides a reasonably accurate description of the energy modification due to the external gas. In agreement with the conjecture of ref. [22], the bulk terms are completely unaffected by the external medium (a m V (ρ) = a V =cst., a m V sym (ρ, δ) = a V sym =cst.), while important effects on the surface properties of nuclei are seen.

The implementation of such an improved energy functional in the extended NSE model is in progress.

Supra-saturation densities and the strangeness phase transition

If it is well admitted that hyperonic and deconfined quark matter could exist in the inner core of neutron stars, a complete understanding of the composition and equation of state of dense matter is far from being achieved. Concerning hyperons, simple energetic considerations suggest that they should be present at high density [24]. However, in the standard picture the opening of hyperon degrees of freedom leads to a considerable softening of the equation of state [24,25,26], which in turns leads to maximum neutron star masses smaller than the observed masses of many neutron stars, in particular the high values obtained in recent observations [1]. This puzzling situation implies that the hyperon-hyperon and hyperon-nucleon couplings must be much more repulsive at high density than presently assumed [27,28,29,30,31,32], and/or that something is missing in the present modelization.

The generic presence of attractive and repulsive couplings suggests the existence, in a modelindependent manner, of a phase transition involving strangeness. Let us consider a simplified situation where matter is uniquely composed of neutrons and Λ-hyperons. The energy density = kin + pot at zero temperature is obtained with the energy density functional proposed by Balberg and Gal [33]

pot (ρ n , ρ Λ ) = 1 2 (a N N + b N N ) ρ 2 n + c N N ρ δ+1 n + 1 2 a ΛΛ ρ 2 Λ + c ΛΛ ρ γ+1 Λ + a Λn ρ n ρ Λ + c Λn ρ n ρ Λ ρ n + ρ Λ [ρ γ n + ρ γ Λ ] , (8)
Due to the attractive part of the nΛ coupling, a mixture of the two particle species admits a bound state at finite density. This rather general feature of the energy density is found within other parameterizations and other models, e.g. the parameterization of the energy density functional from G-matrix calculations by Vidana et al. [34]. The existence of a minimum in the energy functional for symmetric matter means that such a state represents the stable matter phase at zero temperature. On the other side, the vanishing density gas phase is always the stable phase in the limit of infinite temperature. This implies that a dilute-to-dense phase change in strange matter has to be expected on very general grounds. It is intertesting to remark that the minimum, for a certain interval of neutron densities, is also observed as a function of the Λ density. This is again due to the interplay between the attractive Λ -Λ interaction required by hypernuclear data, and the repulsion at high density required by the two-solar mass neutron star observation. This information means that the expected phase change is expected to have strangeness contributing to the order parameter. The presence of a minimum is however not sufficient to discriminate between a smooth crossover and a phase transition. In the next section we will therefore demonstrate the existence of a phase transition by explicitly calculating the nΛ phase diagram.

Neutron-Λ phase diagram

First-order transitions are signaled by an instability or concavity anomaly in the mean-field thermodynamic potential, which has to be cured by means of the Gibbs phase equilibrium construction at the thermodynamic limit. At zero temperature, one thermodynamic potential is given by the total energy

tot (ρ n , ρ Λ ) = pot (ρ n , ρ Λ ) + kin (ρ n , ρ Λ ) + (ρ n m n + ρ Λ m Λ ) c 2 (9)
The spinodal region is then recognized as the locus of negative curvature of the energy surface, C min < 0. The corresponding eigenvector defines a direction in the density space given by

ρ n ρ Λ = C nΛ C min -C nn = C min -C ΛΛ C Λn ( 10
)
with C ij = ∂ 2 e tot /∂ρ i ∂ρ j . This instability direction physically represents the chemical composition of density fluctuations which are spontaneously and exponentially amplified in the unstable region in order to achieve phase separation, and gives the order parameter of the associated phase transition.

The zero temperature instability region of the nΛ mixture is shown in the left part of Fig. 3. We can see that a large portion of the phase diagram is concerned by the instability. We can qualitatively distinguish three regions characterized by different order parameters. Below nuclear saturation density, we observe an isoscalar ρ n ≈ ρ Λ instability, very close to ordinary nuclear liquid-gas, with Λ's playing the role of protons. Increasing neutron density the phase separation progressively changes towards the strangeness ρ S = -ρ Λ direction: the two stable phases connected by the instability have close baryon densities but a very different fraction of Λ's. For high ρ Λ we observe the same behavior with the roles of neutrons and Λ-hyperons exchanged. The part of the phase diagram at high neutron density comprises the region physically explored by supernova and neutron star matter, which is characterized by chemical equilibrium for reactions implying strangeness, µ S = µ n -µ Λ = 0. The strangeness equilibrium trajectory is represented by a dashed line in Fig. 3. This physically corresponds to the sudden opening of strangeness, observed in many modelizations of neutron star matter (see e.g. [33,26]).

Let us stress that the existence of a strangeness phase transition at high density is not a general model-independent feature, although many models show it. There are others, for instance the G-matrix models of ref. [34], which do not show an instability in this region. The reason is essentially due to the absence of a ΛΛ interaction in these microscopic calculations. Owing to the lack of reliable information on the hyperonic interactions, which would discriminate between different models, we cannot affirm the existence of the strangeness phase transition, related to this instability, but, turning the argument around, the presence of this phase transition in a physical system would allow to learn much about the shape of the interaction.

Up to now we have only presented results at zero temperature. The extension to finite temperature, as it is needed for supernova matter, is relatively simple in the mean-field approximation. The appropriate thermodynamic potential at non-zero temperature is given by the Helmholtz free energy

f T (ρ n , ρ Λ ) = tot (ρ n , ρ Λ ) -T s (ρ n , ρ Λ ) ( 11
)
where s is the mean-field entropy density. Chemical potentials can be obtained by differentiating this expression. Then the Gibbs construction is performed from the combined analysis of µ n (ρ n ) at constant µ Λ , and µ Λ (ρ Λ ) at constant µ n . The phase diagram as a function of temperature is presented in the right panel of Fig. 3. This phase diagram exhibits different interesting features. We can see that the three regions that we have tentatively defined at zero temperature appear as distinct phase transitions at finite temperature. The first phase transition (zone II in Fig. 3 -right panel) separates a low density gas phase from a high density more symmetric liquid phase, very similar to ordinary liquid-gas. The second one (zone III in Fig. 3 -right panel) reflects the instability of dense strange matter towards the appearance of neutrons and has an almost symmetric counterpart (zone I in Fig. 3 -right panel) in the instability of dense neutron matter towards the formation of Λ-hyperons. Up to a certain temperature, this latter phase transition is explored by the µ n = µ Λ trajectory, meaning that it is expected to occur in neutron stars and supernova matter. At variance with other known phase transitions in nuclear matter, this transition exists at any temperature and is not limited in density; it is always associated with a critical point, which moves towards high density as the temperature increases. This means that criticality should be observed in hot supernova matter, at a temperature which is estimated as T c = 14.8 MeV in the present schematic model.

The consequences of these findings for the composition of neutron star matter are drawn in Fig. 4. The left panel shows the Λ fraction Y Λ = ρ Λ /(ρ n + ρ Λ ) as a function of the baryon density under the condition µ n = µ Λ . The crossing of the mixed-phase region with increasing neutron density implies that, as soon as the lower density transition border is crossed, the system has to be viewed as a dis-homogeneous mixture of macroscopic regions composed essentially of neutrons, with other macroscopic regions with around 25% of hyperons. The extension of the Λ-rich zone increases with density until the system exits the coexistence zone, and becomes homogeneous again. The density values associated to the phase transition slightly change with the energy functional assumed. To give a simple example, we present on the same Fig. 4 a calculation where Λ's are put in equilibrium with symmetric ρ n = ρ p non-strange matter, that is eq.( 8) is still used, but the symmetry energy in the nucleon sector b N N is put to zero [33]. We can see that only small differences are observed in the strange phase transition considering pure neutron matter or symmetric matter.

On the right panel of Fig. 4 the Λ-densities for the two coexisting phases are displayed as a function of temperature, again under the condition of µ n = µ Λ , physically relevant for neutron star and supernova matter. The extension of the coexistence region decreases with increasing temperature. The latter finally disappears at T c = 14.8 MeV. This well illustrates the fact that the critical point of the strangeness phase transition moves to higher density, crossing the physical line µ n = µ Λ at a given temperature.

Conclusions

In this contribution we have analyzed the phase diagram of stellar matter as it is produced in core-collapse supernovae and proto-neutron stars in the thermodynamical conditions where such matter can be described by nucleonic and hadronic degrees of freedom.

At baryonic densities lower than nuclear saturation, the first-order phase transition of neutral nuclear matter is quenched in stellar matter due to the effect of Coulomb frustration. As a consequence, matter is expected to be clusterized even at finite temperature in the whole subsaturation regime.

The composition and thermodynamics of such clusterized matter has been explicitly worked out in the framework of a quasi-particle model where inter-particle interactions are modeled through excluded volume. Coulomb correlations with the neutralizing electron gas are accounted for in the Wigner-Seitz approximation and residual nuclear correlations are included in an effective way through an in-medium modification of the cluster energy functional with parameters fitted from Hartree-Fock calculations with a realistic effective interaction. The same formalism is used to evaluate the free nucleon self-energies.

The expectation of the phase transition quenching is confirmed by this model, which shows that the single nucleus approximation widely used is a poor approximation at finite temperature and a large distribution of cluster species has to be included in supernova modeling.

In the supersaturation regime, we have used the same mean field approximation with effective phenomenological energy functionals in order to describe the characteristics of the phase diagram in the presence of strangeness.

We have shown that already a simple system composed of neutrons and Λ hyperons in thermal equilibrium presents a complex phase diagram with first and second order phase transitions. Some of these phase transitions are probably never explored in physical systems. However, a possible phase transition at super-saturation baryon densities, from non-strange to strange matter is expected to be observed both in the inner core of neutron star and in the dense regions of core-collapse supernova. For this latter phenomenology, a critical point is predicted and the associated critical opalescence could have an impact on supernova dynamics [5]. The immediate consequence of that is that the opening of hyperon channels at high density should not be viewed as a continuous (though abrupt) increase of strangeness in the matter, observed in many models of hyperonic matter, cf e.g. [26], but rather as the coexistence of hyperon-poor and hyperon-rich macroscopic domains.

An extension of this simple model to include all possible hyperons and resonances is in progress.

Figure 1 .

 1 Figure 1. Left side: Cluster number densities as a function of the cluster size a at different densities (expressed in g/cm 3 ) of the extended NSE model at T = 1.6 MeV and asymmetry chemical potential µ I = 1.68 MeV. Right side: Comparison between canonical (full line) and grand-canonical (dashed line) predictions for the cluster distribution in a specific thermodynamic condition relevant for supernova dynamics. Figure taken from ref. [19].

Figure 2 .

 2 Figure2. Left part: isoscalar and isovector bulk and isoscalar surface term of the in-medium modified mass formula extracted from Hartree-Fock calculations in the Wigner-Seitz cell. Right part: comparison between the microscopic Hartree-Fock volume energy per particle and surface energy per A 2/3 and the parameterization eq.(6) for three isotopic chains as a function of the bulk proton-neutron asymmetry. The model surface term is given by a m S (ρ) -a m Ssym (ρ, δ)δ 0 2 ,

Figure 3 .

 3 Figure 3. Left part: (n, Λ) mixture at T = 0: Borders of the phase coexistence region (red thick lines), constant-µ n paths after Maxwell construction (blue thin lines) and µ n = µ Λ trajectory after Gibbs construction (dark green dotted line). Right part: borders of the phase coexistence region of the nΛ mixture for different temperatures, T =0, 5, 10, 20 MeV. The full circles mark the critical points. Figures taken from ref. [35].

Figure 4 .

 4 Figure 4. Left: Λ fraction as a function of the baryon density within the condition µ n = µ Λ for a nΛ and, respectively, a symmetric ρ n = ρ p npΛ mixture at T =0. The full lines correspond to the stable mean-field results; the dashed lines illustrate the Gibbs constructions; the horizontal lines indicate the relative amount of Λ-hyperons at the frontiers of the phase-coexistence domain. Right: Λ density for the two coexisting phases as a function of temperature for µ S = 0. Figures taken from ref. [35].